Mathematics
Maclaurin Mathematical Olympiad
2023
3

3

ABCD is a square and X is a point on the side DA such that the semicircle with diameter CX touches the side AB.

Find the ratio AX : XD.

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Answer
SOLUTIOn

Since we are interested in the ratio of sides, we can assign a side length for convenience as this would not affect the ratio of lengths. Let CD =2.

Let O be the centre of the semicircle, let Y be the point of tangency on AB and let YO meet CD at Z.

Note that OY = OX = OC as they are all radii. Since OY is perpendicular to AB, OZ is perpendicular to CD. Then Z is the midpoint of DC because triangles COZ and CXD are similar with scale factor 2.

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