Answer to Question #158315 in Geometry for Einstein

Question

Let BCB′C′ be a rectangle, let M be the midpoint of B′C′, and let A be a point on the circumcircle of the rectangle. Let triangle ABC have orthocenter H, and let T be the foot of the perpendicular from H to line AM. Suppose that AM= 2, [ABC] = 2020, and BC= 10. Then AT=m/n, where m and n are positive integers with gcd (m,n) = 1. Compute 100m+n.

Answer 1

$AM\cdot MK=C'M\cdot MB'\implies MK=C'M\cdot MB'/AM$

From properties of orthocenter:

$BM'=CM', HM'=A'M'=AM$ , $AA'$ is the diameter

Then:

$A'K'=AK,AT=HK'$

$AT=AM+MK-2AM=\frac{BC^2}{4AM}-AM$

$AT=\frac{10^2}{4\cdot2}-2=10.5=\frac{21}{2}$

$m=21,n=2$

$100m+n=100\cdot21+2=2102$