# get coordinates to draw a triangle inside another triangle

I am not a Math guy. So, if this question is silly, then please forgive me.

I have a triangle which has the following coordinates: (all the measurements are in cm.)

A - (35, 0)
B - (70, 70)
C - (0, 70)


I have filled this triangle with black color.

Now, I want to draw another triangle inside it. I don't know what the co-ordinates of that triangle will be. But after drawing new triangle i will fill it with red color and it should look like there is a black border of 1 cm around it.

Can you please let me know the co-ordinates for new triangle?

Update:

@Floris gave me this value of 3 pairs of vertices:

$$(70-\frac{1}{2}(\sqrt{5}+1),69)$$, $$(35,\sqrt{5})$$ and $$(\frac{1}{2}(\sqrt{5}+1),69)$$

In this case:

A - (35, 0)

B - (70, 70)

C - (0, 70)

P - $$(35,\sqrt{5})$$

Q - $$(70-\frac{1}{2}(\sqrt{5}+1),69)$$

R - $$(\frac{1}{2}(\sqrt{5}+1),69)$$

Which gives me output like:

But now suppose I want to have border of 10 cm:

So, I changed the value to: (please consider x = 10)

A - (35, 0)

B - (70, 70)

C - (0, 70)

P - 35,x√5

Q - 70-((x/2)(√5+x)),70-x

R - (1/2)(√5+x),70-x

But I am getting

I want the formula for inner triangle's vertices. In short I need P, Q, and R having some x.

Note that the edges are given by $$\overline{AB}=\{(x+35,2x):x\in[0,35]\}$$, $$\overline{BC}=\{(x,70):x\in[0,70]\}$$ and $$\overline{AC}=\{(35-x,2x):x\in[0,35]\}$$. The linepieces on centimetre from the edges are given by $$\hat{AB}=\{(35+x-\frac{2}{5}\sqrt{5},2x+\frac{1}{5}\sqrt{5}):x\in[0,35]\},$$ $$\hat{BC}=\{(x,69):x\in[0,70]\}$$ and $$\hat{AC}=\{(35-x+\frac{2}{5}\sqrt{5},2x+\frac{1}{5}\sqrt{5}):x\in[0,35]\}$$ which have intersections $$(70-\frac{1}{2}(\sqrt{5}+1),69)$$, $$(35,\sqrt{5})$$ and $$(\frac{1}{2}(\sqrt{5}+1),69)$$, which are the coordinates of the red triangle.
This process can be generalised for different borders. Note that $$\hat{AB}=\overline{AB}-(\frac{2}{5}\sqrt{5},\frac{1}{5}\sqrt{5})$$ here $$(\frac{2}{5}\sqrt{5},\frac{1}{5}\sqrt{5})$$ is a unit vector which is orthogonal to $$\overline{AB}$$. So if we want a border of length $$l$$, then we can just take $$\hat{AB}_{l}=\overline{AB}-l\cdot(\frac{2}{5}\sqrt{5},\frac{1}{5}\sqrt{5})$$ This gives us $$\hat{AB}_{l}=\{(35+x-\frac{2l}{5}\sqrt{5},2x+\frac{l}{5}\sqrt{5}):x\in[0,35]\},$$ $$\hat{BC}_{l}=\{(x,70-l):x\in[0,70]\}$$ and $$\hat{AC}_{l}=\{(35-x+\frac{2l}{5}\sqrt{5},2x+\frac{l}{5}\sqrt{5}):x\in[0,35]\}.$$ To calculate the coordinates of the new triangle we have to find the intersections of the triangles, these are given by: $$\hat{AB}_{l}\cap\hat{AC}_{l}=\{(35,l\sqrt{5})\}$$ as due to symmetry $$35=35-x+\frac{2l}{5}\sqrt{5}\Leftrightarrow x=\frac{2l}{5}\sqrt{5}$$. From which it then follows that $$2x+\frac{l}{5}\sqrt{5}=l\sqrt{5}$$. For $$\hat{AB}_{l}\cap\hat{BC}_{l}$$ we can solve $$70-l=2x+\frac{l}{5}\sqrt{5}\Leftrightarrow x=35-\frac{l}{2}(1+\frac{1}{5}\sqrt{5})$$ which then gives $$35+x-\frac{2l}{5}\sqrt{5}=70-\frac{l}{2}(1+\sqrt{5})$$. Hence $$\hat{AB}_{l}\cap\hat{BC}_{l}=\{(70-\frac{l}{2}(1+\sqrt{5}),70-l)\}.$$ Finally by symmetry we find that $$\hat{AC}_{l}\cap\hat{BC}_{l}=\{(\frac{l}{2}(1+\sqrt{5}),70-l)\}.$$
• First of all thanks for giving me an accurate answer. I have checked (70−12(5‾√+1),69) , (35,5‾√) and (12(5‾√+1),69), and it works perfectly. I really don't understand the calculations that you have put in your answer. And now I need a border of 2 or 3 pixels at some places. So, can you please give me the formula for border = x? Sorry, if I asked for too much help. – Vishal Mar 29 '19 at 7:14
• @Vishal I've changed the bounds of $x$ to precisely give the edges of the red triangle. – Floris Claassens Mar 29 '19 at 10:03