geometrytriangles
I have a right triangle with two 45degree angles.
I know the points for the two coordinates opposite the right angle. I need to calculate the missing point.
I have seen similar questions here, but I have not understood the answers. There were lots of symbols and fancy words that made no sense to me. I tried to do some calculations, but my results are not quite right.
I'm hoping someone can provide me with a solution using layman's terms and a straight-forward formula.
Thanks very much
In Elements I, 32, Euclid proves the angles of a triangle sum to "two right angles" ($180^o$) by first showing that an exterior angle equals the sum of the two opposite interior angles. So he can't use the former to prove the latter.
Instead, he shows that exterior $\angle ACD=\angle ABC+\angle BAC$ by drawing$$CE\parallel BA$$and observing that, by I, 29, alternate interior$$\angle BAC=\angle ACE$$and corresponding$$\angle ABC=\angle ECD$$Therefore, the whole exterior $\angle ACD=\angle BAC+\angle ABC$.
Only then does Euclid prove--almost as an afterthought--the theorem fundamental in his geometry, that the angles of a triangle sum to $180^o$, or "two right angles" as he puts it.
Best Answer
Let $A=(x_A,y_A)$ and $B=(x_B,y_B)$ be the given points. Let their midpoint be $M=((x_A+x_B)/2,(y_A+y_B)/2)$ and $X=(x,y)$ the point to be found.
Then you have that $MX$ is perpendicular to $AB$ and that $MX=AM$. This gives you two equations, to find $x$ and $y$.