In a right-angled triangle ABC (∠C=90°), the height CD equals to h is drawn.
The points M and N are chosen on the continuation of AB such that NA=AD and DB=BM (Fig. 1).
Find the distance from the point C to the orthocenter of the triangle CMN (Fig. 2).
Steftcho P. Dokov
The unexpected answer "3*h ..." came from vectors manipulations and a dot product. Now, a geometric solution can be "adjusted" to the known answer.
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Hui Liu
Take a Cartesian coordinate: let C be the original, DM and DC be the x-axis and y-axis. Let AD=a, DB=b, CQ=x, then C=(0, 0), D=(0,-h), B=(b,-h),A=(-a,-h), etc.. From AC perpendicular to BC, we have h^2=ab; and from CN perpendicular to QM, we have -4ab+h^2+hx=0. Now we can see that
x=3h.
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