Are there nine?

Geometry Level 4

A $\triangle \space ABC$ with its sides $AB, BC \space and \space CA$ having a slope of

${m}_{BA}=\tan 15^\circ$
${m}_{BC}=\tan 75^\circ$
${m}_{CA}=\tan 135^\circ$

While constructing the nine point circle , how many points will the circle have in common with the triangle.?

Clarification: Having a point in common means that the circumference of the circle and the three sides of the triangle are intersecting (and/or) touching each other.

This question is part of the set Geometry problems .

The answer is 3.

1 solution

Abhay Tiwari
Jun 12, 2016

The slopes of the sides of the triangle mentioned above will form an equilateral triangle as shown below:

And for an equilateral triangle the circumference of the nine point circle will touch only at three points, as mid point of the three sides will be the same as the feet of the altitude of three sides respectively. And so, there are only three points which the circle and the triangle have in common.

Are there nine?

This question is part of the set Geometry problems .

The answer is 3.

1 solution

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