Alien mothership attacks dome city

Two semicircles with parallel bases. Show that HF and GE intersect on the perpendicular line to AC above A.

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Solution by Ahmet ÇETİN.

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One reply on “Alien mothership attacks dome city”

Proof 1. To show dotted line works, we just need to show that the ratio of the split of h between green and purple regions is the same as x/y.
But, by similar right triangles, x/h = a/c, and y/h = d/b, so x/y = ab/cd. But, the area argument in the “Sailing by Night” problem shows that the h split ratio is just ab/cd, since the angle between a and b is 180 minus the angle between c and d.

Proof 2. Once the middle circle is drawn, you can also see that it is a special case of Pascal’s theorem, applied to the hexagon HFCCAA.
The AA side is the horizontal tangent at the bottom, labeled y, and the CC side is labeled x. Pascal says the three intersections HF^CA (= middle pt) FC^AA (=E) and CC^AH (= G) are collinear.

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