# Finding the middle coordintes of each side of an equilateral triangle

I have a problem with finding the center coordinates of each side in an equilateral triangle. I've linked an image below that shows exactly which coordinates I'm after.

I understand that I can work out the height of then triangle by using pythagoras theorem. I can split the triangle down the middle and calculate the opposite of one of the triangles.

However, I can't figure out how to work out the coordinates marked with a "?"

Is there any formula I can use to work this out? The center part of the triangle would be it's origin at (0,0) and the sides would be 3 units each.

Thankyou • Do you know about the properties of an equilateral triangle and the chords connecting the vertices with the opposite side through the center? Check out this link. – John Wayland Bales Jul 16 '19 at 19:32
• Your image is incorrect, so it confuses you - near the question marks must be right angles. – MarianD Jul 16 '19 at 19:38

The height of the equilateral triangle is $$h={\sqrt 3 \over 2}a$$

(you have $$a=3$$), and coordinates of its vertices are

$$A = \left(-\frac a 2, -\frac h 3\right),\quad B = \left(\frac a 2, -\frac h 3\right),\quad C = \left(0, {\frac 2 3}h\right).$$ The center of AC is $${A+C \over 2}$$, the center of BC is $${B+C \over 2}$$.

• This explains it excellently. Thankyou very much MarianD! – juicy89 Jul 16 '19 at 20:07
• @juicy89, you're welcome. – MarianD Jul 16 '19 at 20:16

Hint: Draw a line parallel to the base through the centre. Then drop a perpendicular from one of the $$?$$-points to meet this line. You have a right triangle whose legs are the coordinates of that point.

To calculate these legs, use parallelisms to determine that this little triangle is a $$30$$-$$60$$-$$90$$ one, so that it is similar to one-half of the original. Then Bob's your uncle!

Assuming one of the vertices is on axis $$y$$, it's coordinates will be A$$(0,\sqrt{3})$$ (use Pythagoras and $$30-60-90$$ triangle). The bottom side will lie on the line $$y=-0.5\sqrt{3}$$. Thus, the coordinates of the other two vertices are: $$B(-1.5, -0.5\sqrt{3})$$ and $$C(1.5, -0.5\sqrt{3})$$. To find coordinates of a midpoint of a segment, you simply take arithmetic mean of the corresponding coordinates of the end points.