Concyclicity in Two Semicircles

What Might This Be About?

28 December 2013, Created with GeoGebra

Problem

Point $C$ is taken on segment $AB;$ form two semicircles with centers $F$ and $G$ on $AC$ and, respectively, $BC$ as diameters. Let $DE$ be their common tangent.

Concyclicity in Two Semicircles - problem

Prove that $A,$ $D,$ $E,$ and $B$ are concyclic.

Hint

This configuration has already been considered elsewhere. The right triangles $ADC,$ $CEB,$ $DCE$ are similar which lays a way of computing the angles $ADE$ and $DEB.$ These can be used to prove that the quadrilateral $ABED$ is cyclic.

Solution

Connect $AD,$ $CD,$ $CE,$ $BE.$ Observe that inscribed $\angle CAD$ equals $\angle CDE$ formed in circle $(F)$ by chord $DC$ and the tangent at $D.$ Similarly, $\angle CBE=\angle CED.$

Concyclicity in Two Semicircles - solution

So there are three similar right triangles $ADC,$ $CEB,$DCE.$ Further,

$\begin{align} \angle ADE + \angle ABE &= (90^{\circ}+\angle CDE) + \angle ABE \\ &= 90^{\circ}+(\angle BCE + \angle ABE) \\ &= 90^{\circ}+90^{\circ}=180^{\circ}, \end{align}$

implying that the quadrilateral $ABED$ is indeed cyclic.

In passing, note that the center $J$ of $(ABED)$ lies on the intersection of the perpendicular bisectors of $AB,$ $AD,$ and $BE$ (as these are also chords in $(ABED)$) and the semicircle on $FG$ (because $\Delta FGJ$ is right (and similar to the above mentioned right triangles.)

Acknowledgment

The problem is an auxiliary lemma from a bigger work by Emmanuel Antonio José García. Antonio also observed that the proof never uses the fact that the semicircles are touching. Hence, two additional cases:

Concyclicity in Two Semicircles - general case 1

Concyclicity in Two Semicircles - general case 2

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