A right triangle is divided in two triangles by an altitude. The three incircles are shown with three tangency points. Prove that the two red line segments are congruent.
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Seeing double II
Tag: circle
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Sunrise over green mountain
An acute triangle mountain ABC has altitudes BE and CF. The dotted tangent lines to the sunny circumcircle (AEF) at E and F intersect at a point M. Show that M is on the mountain’s base BC.
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The silly walk
A circle and several triangles. Prove that the green triangle is isosceles.
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The sun hat
Two triangles share a circumcircle and vertex, with one edge of the orange triangle containing the feet of two of the altitudes of the blue triangle. Show that the orange triangle is isosceles.
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Uneasy the head
The green zigzag crown segments would extend through either B or C. Show that the arcs along the top are equally spaced.
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Falling in
Start with an acute triangle and form a new triangle from the points of tangency of its inscribed circle. Continue this process to make make the triangle with blue vertices. What is the maximum possible angle at a blue vertex?
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Blue convergence
Points B, C, D are on a circle with centre O and diameter COC’. Point E is on the line BC such that DE is perpendicular to COC’. Show that the perpendicular bisectors of EB and ED and the line DC’ are concurrent.
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The bike chain II
Two circles and four coloured common tangents. Prove they are congruent.
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The marble track
A triangle with two altitudes and a circle centred in their intersection. Prove that the red line segments are parallel.
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Bottoming out II
A semicircle with three squares. What’s the angle α?