A semicircle with a radius, a chord and an altitude. What is the angle?

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## Solution

The angle α can be anything between 0° and 90°.

First, let me admit that I thought the answer would be 60°. This, however, is not a unique solution. Let’s see where I went wrong.

First, triangles ADC and AEB are congruent because of AAS: they share α and a right angle. Because of reflection symmetry in the radius, triangles AEB and AEF are congruent.

The last step would be to show that AEF and AGF are congruent, in which case β would equal α and 3α would equal 180°. This last step however is not allowed in general, since GF is not congruent with EF.

Ignacio Larrosa Cañestro has made an animation showing the continuum of solutions clearly.

## Poem

Looking a point in the sky

The astronom counts the stars

Talking to the moon

And the planets with his heart

Day is going, night is coming

Heaven is open

Giving it’s beauty

A work of diamonds in progress

Tangling planets

Spatial dance

The sun is sleeping

## One reply on “The observatory dome”

Figures are sometimes misleading, one must be very careful. But we must not fall into the Bourbaki’s hyperreaction of completely banishing them from mathematics …