All Triangles are Isosceles // 203

Or is there?

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Answer on page 296

All Triangles are Isosceles This puzzle requires some knowledge of Euclidean geometry, which nowadays isn't taught . . . Ho hum. It's still accessible if you're prepared to take a few facts on trust.

An isosceles triangle has two sides equal. (The third could also be equal: this makes the triangle equilateral, but it still counts as isosceles too.) Since it is easy to draw triangles with all three sides different, the title of this section is clearly false. Nevertheless, here is a geometric proof that it is true.

This triangle is isosceles ­

except that it clearly isn't.

(1) Take any triangle ABC.

(2) Draw a line CX that cuts the top angle in half, so that angles a and b are equal. Draw a line MX at right angles to the bottom edge at its midpoint, so that AM = MB. This meets the previous line, CX, somewhere inside the triangle at the point X.

(3) Draw lines from X to the other two corners A and B. Draw XD and XE to make angles c, d, e and f all right angles.

(4) Triangles CXD and CXE are congruent ­ that is, they have the same shape and size (though one is the other flipped over). The

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