The center of the circle passing through the midpoints of the sides of am isosceles triangle ABC lies on the circumcircle of triangle ABC. If the larger angle of triangle ABC is α and the smaller one β then what is the value of α - β?
If the center of the circle is on the circumcircle and the circle itself passes through all three of the midpoints, that center must be located at the apex of the isosceles triangle.
Call the distance of the center to the midpoint of the base 1 unit. The distance to each other midpoint must also be 1, making the continuation of the other side of the side to the base also have length 1, so the equal sides have length 2 each. So one half of the isosceles triangle that forms a right triangle has a leg of length 1 and hypotenuse 2. The base angle is 30° and the apex angle is therefore 180 - 2*30 = 120° and the difference between them is 90°.
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Posted by Charlie
on 2024-08-15 09:17:10 |
proposed solution
