Prove that angles in a circle equal each other

Question

Let a circle in the Euclidean plane be given, let $AB$ be a diameter, and let $CD$ be the tangent through point $A$. Let $E$ and $F$ be two points on the circle, on the same side of $AB$ as $C$, and with $F$ between $E$ and $A$.

Show that angle$AFE$= angle$DAE$

$\textbf{Hint: }$ They have the same arc $ABE$ – L F May 10 '15 at 16:38 — L F, May 10 '15 at 16:38
Search for "angles in the alternate segment". – Mick May 10 '15 at 17:06 — Mick, May 10 '15 at 17:06

score 1 · Accepted Answer · edited Jun 12 '20 at 10:38

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Let $\angle DAE=\alpha$, then $\angle EAC= 180-\alpha$, so the arc $AFE= 2(180-\alpha)$, so the arc $EBA=360-AFE=2\alpha$, then $2\alpha=EBA=2\angle AFE$ so $\angle AFE=\alpha=\angle DAE$

$\textbf{Note:}$ $\angle DAF =\alpha$ (green shadow) and $\angle FAC= 180-\alpha$ (black shadow)

edited Jun 12 '20 at 10:38

Community

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answered May 10 '15 at 16:41

L F

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The shaded green angle at $A$ appears to be $\angle DAF$, not $\angle DAE$. – Barry Cipra May 10 '15 at 17:36
I shaded FAC, I don't know ho change size of angle in geogebra. (I've exported imagen from there) – L F May 10 '15 at 17:37
I mean the light green shading, not the dark shading. – Barry Cipra May 10 '15 at 17:39
I'll do the corrections in text, because I deleted the source of image. – L F May 10 '15 at 17:40

Prove that angles in a circle equal each other

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