Kenmotu & Circles Relationship

Geometry Level 3

The above is an equilateral triangle inside of which three identical squares are intersecting at the point named first Kenmotu point .

True or False?

If the triangle is equilateral, then the first Kenmotu point is the center of the incircle .

If the first Kenmotu point is the center of the incircle, then the triangle is equilateral.

Note: Before you try out this problem, read this article .

This is the easy version of the previous problem .

Both statements are true One statement is true, one statement is false Both statements are false

1 solution

Nick Kent
Jan 23, 2019

Let's look at the isosceles triangles formed by two sides of adjacent squares and one triangle side with first Kenmotu point on top. If first Kenmotu point is incenter, then the heights of the isosceles triangles are equal to incircle radius and are all the same (since the incircle touches their bases). The legs are also equals because they are sides of three identical squares. Thus, all three triangles are equal: they are isosceles and their legs and heights are equal.

Since there are squares between them the angles on top are equal $(2\pi-3(\pi/2))/3=\pi/6$ . The angles at the bases are equal $(\pi-\pi/6)/2=5\pi/12$ , the angles at the bases outside of the triangles equal $\pi-5\pi/12$ . Then the angles of the initial triangle equal $2\pi-2(\pi-5\pi/12)-\pi/2=\pi/3$ . Thus, the initial triangle is equilateral.

Kenmotu & Circles Relationship

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