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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!

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    It would be better if you posted the options. – 2017-01-22
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    @N.S.JOHN Added. – 2017-01-22

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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.

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    Hi, thank you for your detailed answer! "(by computing dot products) the given options form a orthocentric system." -> could you please provide the name/reference url for this technique where I can study it more? "The orthocenter of ABC is the incenter of the orthic triangle." -> is this a property valid for any triangle. – 2017-01-23
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    @GaurangTandon: http://mathworld.wolfram.com/OrthocentricSystem.html . And yes, the orthocenter is always the incenter of the orthic triangle, it is enough to apply angle-chasing. – 2017-01-23