Does the incenter divide the triangle into a ratio of 2:1?

Question

I'm trying to get the radius of the circle.

First I use the Pythagorean theorem $13^2=5^2+x^2$ and $x=12$. Then I know the height of the triangle. To get the radius i use the Pythagorean theorem $r^2=5^2+(\frac{1}{3}\cdot12)^2$.

But here comes the problem, isn't the red part $\frac{1}{3}$ of the height, as the ratio of the incircle is $2:1$? My classmates don't agree, so who's right?

The triangle:

Edit: Is the black dot in the triangle the incenter or centroid?

Answer 1

The circumradius R can be found (if we let a,b,c be the sides and A=area) with the formula $$R=\frac{abc}{4A}=\frac{10\cdot13\cdot13}{4(5\cdot12)} =\frac{1690}{240}\approx 7.041666667$$ The "red" part of your diagram is then $\quad 12-7.041666667=4.958333333\quad$ so the ratio is more like $\space7:5\space$ than $\space2:1.\space$ The eye views proportions incorrectly, often making lower "heights" appear greater than they are.

Here and here and here are links discussing the circumcircle radius you seek and below is a picture.