Question:
The coordinates of feet of altitudes from the vertices of a triangle on the opposite sides are $(20, 25)$, $(8, 16)$ and $(8, 9)$. The coordinates of the vertex of the triangle are:
(four multiple choice options out of which 3 have to be selected)
The options:
$(5, 10)$; $(50, -5)$; $(15, 30)$; $(10, 15)$
My attempt:
I know very well that I can assume any two vertices to be $(h, k)$ and $(p, q)$ and then use $m_1\cdot m_2=-1$ for perpendicular lines to get multiple equations and then solve them.
However, this is an objective question and hence needs a shorter and simpler approach for which I need hints.
Please don't suggest checking the options. That would mean checking 4 combinations (ABC, ABD, ACD, BCD). I think that there could be a shorter approach for this question instead of reverse checking the options.
Thank you!
Best Answer
The orthocenter of $ABC$ is the incenter of the orthic triangle.
If you check that $(10,15)$ is the incenter of the orthic triangle (or simply notice it is the only point lying inside the orthic triangle) it follows that $(5,10),(15,30),(50,-5)$ are the vertices of the original triangle, also since (by computing dot products) the given options form a orthocentric system.