How to teach a kid geometry I found my kid more interested in shapes than numbers, adding 14 and 7 to get 21 is boring, while triangle, rectangle and heart shape is fun.
But what could I introduce, besides the basic shapes, e.g. triangle, rectangle, parallelogram, diamond, circle, line, and angle?
I couldn't really go on to Elements; that will kill the interest. Any suggestion on how to keep the introduction basic and fun and still encourage some thinking?
 A: I highly recommend the Keys to Geometry workbooks, which are how I fell in love with geometry and construction myself as a kid.
They require no preparation at all.  If your kid can draw a straight line with a ruler (straightedge), he/she is ready.
The books are almost entirely interactive.  Get a good compass, and a straightedge.  (A ruler is okay only if your kid understands not to use the markings on it at all.)  By book seven your kid will be able to construct a perfect regular pentagon on a blank sheet of paper, unassisted.
I have also supervised several students through them (graded their exercises) and I recommend reading the material in the teacher's manual, if possible.  (Or you can just get the workbooks; you'll be able to check the work fairly easily.)  The teacher's manual contains a section on how to instruct beginners to use a straightedge correctly and precisely, how to hold a compass, etc.—which is likely easy for you to do but is more difficult to put in words so that a child can actually learn it based on your description.
I will also note that it is important that the work be checked for precision.  Don't tolerate sloppiness even a little bit, because if the ability to do the instruction "Draw an arc with center X" is missed (slightly off from X, or a slightly imperfect arc) in book 1, then the later books will be impossible.  These books have extremely little repetition.  That's one of the reasons I loved them so much.

These books very definitely encourage thinking.
I've seen teachers of higher math criticize them because they don't contain actual proofs.  Instead, they have the student work through some concrete examples ("draw a triangle"; "construct the perpendicular bisector of each side"; "do they meet in one point?") and after working through several such examples, ask the student to choose the right word: "The perpendicular bisectors of the sides of a triangle (always/sometimes/never) meet at one point."  All basic rules are demonstrated in such a fashion.
"That's not a proof!" some say.  But this is missing the forest for the trees; other such examples where the answer is "sometimes" are conclusively proven by example, and having the student find the answer for himself or herself is far, far better than reciting a litany of "rules" and then claiming this "proves" something.
(Newsflash for higher math teachers: the only purpose for a proof is to make someone realize something true.  If you convince only yourself but not the student, you have failed.  Okay, off my soapbox.)

Point being, this is an accessible approach to the elements, that kids as young as 8 can do.  And high school students benefit as well.  (Most modern geometry texts have far more text telling the student what is true, and very little requirement that the student work it out himself or herself with an actual compass.)
A: I know you asked about geometry specifically, but it is not the content but the logical reasoning that counts. Find puzzles and recreational math/puzzle books! Puzzles can include tangrams and metal puzzles. Books might have an assortment of interesting stuff like symmetries (you can play with mirrors), tessellations, the Mobius strip and variants, spirographs, fractals, packing problems, and more. Most of these can be fun for little kids and there is no real need to bother with so much 'mathematics' at this stage. Just playing with those stuff is good enough!
A: Puzzles are an excellent way in invoking interest in math. I would look up geometry puzzles online and give him/her some to work out. Increase the difficulty of the puzzles as your kid gets more of an understanding of geometry and along the way, you can introduce geometric concepts that would help in solving the puzzles. And because those concepts would've helped him/her in solving the puzzles, they'll be more likely to be remembered. Better to teach your kid the concepts yourself through problem solving than to have some teacher in the future try to just cram seemingly irrelevant junk into their mind. That's just my theory though. 
A: Some other ideas:


*

*Children at that age have often problems with perspective. Teach him how to draw using lines that go out from the same radiant. Show him how to draw parallelogram using that technique.

*Show him (and/or teach him how to draw) some optical illusions like impossible triangle or elephant with 5 legs.

*Show him how to construct a kaleidoscope using two mirrors. Later it will be easier for him to understand concept of central symmetry.

*Buy him and/or show him how to make on his own some folding books.

*Show him how to make basic polyhedrons or other shapes using stiff wires.
A: While taking him out for a walk let him see shapes in buildings, parks, nature etc. Encourage him to draw at first what he saw after his verbal description of geometrical figures, to the extent he can. You should yourself never draw, but only correct/refine his drawings and and related narrations.
A: I've assisted in teaching kids aged 10–14 mathematical art, and found a lot of success with the following project:
Invent a polyhedron and try to make it! Basically, take construction paper, and have them use straightedge and compass to make shapes and subsequently cut them out. The most "appealing" are regular shapes. Since your child is 4, you should probably cut the shapes for them.
Here were some especially nice moments:


*

*One student tries to make a vertex with $6$ equilateral triangles and asks me why it wasn't working. 

*Multiple students worked on constructing pentagons, which  was a bit of a challenge (for me too!)

*Some students finished early and started coloring their polyhedra, with $4$ or $5$ colors. Of course I bothered them by asking if they could have done it with fewer.

*Some students couldn't "See" the shape, so we would use projections to form planar graphs, or use "nets" to help. This was often a bit of a brain cramp but a lot of fun.
here is a blog post chronicling a collection of activities we did (some were more advanced than others: triangle inequalities, pythagorean theorem, finding a way to approximate pi with arbitrary precision etc.) that might have some nice ideas for you!
A: If it were me, (assuming a child in elementary school) I would start heading into drawing with a ruler and compass.
Working on accurately measuring lengths and angles, drawing circles, lines, and perhaps other shapes, depending on the tools you add, would introduce a visual-art aspect that might help fuel further interest.
For instance, you could work on constructing an equilateral triangle, then adding a few iterations to make it a Sierpinski triangle.
Or you could construct the various centers on a triangle and show they're collinear. 
Or you can work on drawing some simple fractal patterns.
Color and pattern goes hand in hand with art and mathematics.
Origami is also a good way to go, and is a good option for children who can't be trusted with sharp implements. There's a book called "Geometric exercises in paper folding" by T. Sundara Row that was passable, but I imagine there's newer and better books like that out now.
A: It's Not Too Soon to Inspire Geometric Thinking
As they manipulate shapes in daily play and become increasingly aware of spatial relationships throughout their day, preschoolers are actively developing early concepts of geometry. At the preschool level, children learn to recognize geometric shapes by viewing and identifying them as a whole object. Providing children with opportunities to explore and experiment with shapes and their properties allows them to move to the next stage of geometric thinking: understanding the individual characteristics of each shape (for example, a triangle has three sides).
Make Geometric Mastery a Game
Here are some easy ideas on turning playtime into math time!
Block play is great for shape recognition
For example, the simple process of matching and sorting similar blocks at clean up time builds geometric thinking. During block play, pose open-ended questions that spur and advance your child’s geometric thinking. Engage your child with questions like:
How is your block tower different from mine?
What will happen if I take out the bottom block?
Can you tell me what I need to do to make my block building look just like yours?
Can you put 2 blocks together and make another shape?
We are getting ready to clean up now - how will you remember what you built?
Make spatial vocabulary a regular part of conversational language
Acting out stories such as "The Three Billy Goats Gruff," for example, offers numerous opportunities for children to show their understanding of spatial concepts like: under, over, across, near, far.
Turn geometric awareness into a game
Increasing your child's awareness of shapes all around him is as easy as incorporating shape conversations into your playtime. Keep shape blocks or cardboard or paper shapes on hand, and ask questions like:
Have you seen this shape before?
Where have you seen it?
Can you find a shape like this in our home?
Do you think this shape would roll or slide?
Can we stack these shapes?
Can you cut this paper to make another shape?
Can you make a square (circle/triangle) with pipe cleaners/yarn
Resources:
Copley, J. V. (2000).  The Young Child and Mathematics.  Washington, DC:  National
Association for the Education of Young Children.
van Hiele, P. M. (1986).  Structure and Insight:  A theory of mathematics education.  Orlando, FL:  Academic.
A: You can use Origami. 
It will allow your kid to learn from basics (angles, congruences, triangles, etc) to high geometry 
I learned with origami most of basical principles of geometry since kid.
Also you can check https://pdfs.semanticscholar.org/1c1a/397eb31a69dfd2671cb61326491115b779d0.pdf

A: I don't know the solution to this problem. All this is just an idea that researchers should take with a grain of salt if they read it. Different students probably learn different ways. For some students, it might be better not to introduce geometry at all in elementry school and for other students, it may be better to introduce geometry. I believe I wasn't introduced to any geometry at all in elementry school and I'm glad of that because otherwise, my brain might have adapted to not think for itself later and overrely on what I got taught before and not know how to check whether what I was taught before is actually true. If we could change the job market in a way that allows it to work, I think that best education system would use a student centered approach and teach less. That way the teacher doesn't move on from a topic before their students have the time they need to learn and understand it really well. Also that would allow students to discover what they're interested in discovering based on what they were taught which they may be really good at.
Although by now, I already figured out or got told so many ways to derive the distance formula in $\mathbb{R}^2$ from intuitive properties of distance, once when I was a kid at a time that I think was probably before I got taught any geometry, I believe I thought of one and only one way to construct a circle mentally or something like that, and it was to draw a path that is everywhere perpendicular to its direction from the center. It was almost the same method but not quite the same method as the method of using the differential equations for $\sin$ and $\cos$.
For those students who are interested in geometry, one possible idea might be to construct the unit circle on graph paper with a centimeter spacing between the squares. The student may also have to be introduced to the concept of the ordered pair representation of points in a plane even if they have trouble paying attention to that concept because it will be useful later to help them learn about geometry. Next the student could be taught that that the circle can be defined as the path of a particle whose velocity is its position rotated 90° counterclocwise and starts at the point (1, 0) at time 0 although we will not yet call that point that and will just visually show the starting point. Now that the student will see an application for the coordinate representation when they start getting introduced to the differential equations for $\sin$ and $\cos$


*

*$\cos(0) = 1$

*$\sin(0) = 0$

*$\cos' = -\sin$

*$\sin' = \cos$
it may be easier for them to pay attention. Maybe they can be taught how to reduce the equation that describes velocity in terms of position to the 3rd and 4th of those 4 differential equations and reduce the equation that describes the point at time 0 to the 1st and 2nd of those differential equations. After that, it may be better not to be taught more if they don't want to. It's quite likely that they will want to be finished getting taught for now and be left to play and discover what they are interested in discovering. Eventually, they might discover the Pythagorean identity all on their own. This question has an idea similar to the one I proposed here and it has a score of 12 as of the time I posted this answer for the first time.
Technically, without any assumptions about what properties the distance formula satisfies, you can't prove anything about what the distance formula is so I decided to prove in this answer that $d((x, y), (z, w)) = \sqrt{(z - x)^2 + (w - y)^2}$ is the unique binary function from $\mathbb{R}^2$ to $\mathbb{R}$ satisfying the following properties


*

*$\forall x \in \mathbb{R}\forall y \in \mathbb{R}\forall z \in \mathbb{R}\forall w \in \mathbb{R}d((x, y), (x + z, y + w)) = d((0, 0), (z, w))$

*$\forall x \in \mathbb{R}\forall y \in \mathbb{R}\forall z \in \mathbb{R}\forall w \in \mathbb{R}d((x, y), (z, w))$ is nonnegative

*$\forall x \in \mathbb{R} - \mathbb{R}^-d((0, 0), (x, 0)) = x$

*$\forall x \in \mathbb{R}\forall y \in \mathbb{R}d((0, 0), (x, -y)) = d((0, 0), (x, y))$

*$\forall x \in \mathbb{R}\forall y \in \mathbb{R}\forall z \in \mathbb{R}\forall w \in \mathbb{R}d((0, 0), (xz - yw, xw + yz)) = d((0, 0), (x, y))d((0, 0), (z, w))$
and it also satisfies the additional properties


*The area of any square is the square of the length of its edges

*$\forall x \in \mathbb{R}d((0, 0), (\cos(x), \sin(x))) = 1$
I think most young students will prefer to be taught using one of the methods they would like being taught using which is to just take for granted that the distance formula satisfies the following properties


*

*$\forall x \in \mathbb{R}\forall y \in \mathbb{R}\forall z \in \mathbb{R}\forall w \in \mathbb{R}d((0, 0), (z, w)) = d((x, y), (x + z, y + w))$

*$\forall x \in \mathbb{R}\forall y \in \mathbb{R} - \mathbb{R}^-d((0, 0), (y\cos(x) ,y\sin(x))) = y$
and I think for those students, that's the way the teacher should do it. Some of them when they're older might think of the idea to actually check whether there exists a function that satisfies all 7 intuitive properties of distance that I listed earlier and if they're not yet finished school by then, maybe the teacher could talk about the topic of actually checking that and say something like "You're right, I did not yet prove to you that a way of defining distance that satisfies all 7 properties exists."
A: Sometimes kids learn something better when you get them interested in it. I'm guessing they tend to get excited by rare and unusual things. Maybe the teacher could play only the part of the YouTube video Visualizing the sphere and the hyperbolic plane: five projections of each with the orthographic projection of the sphere, muted. After that, they could then bring to the attention of the students that that is a tiling that is not possible on a flat plane. Then they might get all fascinated about the fact that a sphere has different geometry than a flat plane. However, it's probably just spherical geometry and not Euclidean geometry that that would get the students interested in at first. By muting the video, the student hears only the teacher's explanation and not the explanation in the video and can therefore concentrate better on what the teacher is saying.
