Geometry Problem 1502: Right Triangle, Incircle, Inradius, Geometric Mean of 2 Inradii, Angle Bisector, Perpendicular, Tangential Quadrilateral

In the figure, D is the incenter of right triangle ABC, \(e\) and \(f\) are the inradii of the tangential quadrilateral AGDH and CMDH, respectively. Prove that \(BD=2\cdot\sqrt {e\cdot f}\)
Right Triangle, Incircle, Inradius, Geometric Mean of 2 Inradii, Angle Bisector, Perpendicular

Definitions

  • Inradius (plural inradii) is the radius of an incircle.
  • Incircle is a circle inscribed within another shape touching every side of it at one point.
  • Incenter is the center point of an incircle.
  • Tangential quadrilateral is a convex quadrilateral whose sides all can be tangent to an incircle

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Geometry Problem 1502 Solution(s)