Çözüldü Trigonometry - Triangle's Orthocentre (8th Grade on this site)

In an ABC triangle, if the line segment through the vertex B vertically crosses the edge [AC] at the point D, and the distance between B and the triangle's orthocentre H equals the distance between H and D, what is the product of the tangent values of the angles ∡A and ∡C?

Note: Drawing is homework for the interested students.

|BH| = |DH| = a units
tan(90° - ∡A) = a / |CD| ⇒ cot(∡A) = a / |CD| ⇒ tan(∡A) = |CD| / a....(I)
tan(90° - ∡C) = |CD| / (2a) ⇒ cot(∡C) = |CD| / (2a) ⇒ tan(∡C) = 2a / |CD....(II)
Multiplying the equations (I) and (II) side by side and simplifying yields tan(∡A)·tan(∡C) = 2.