Points in a triangle

Geometry Level 3

There are $10$ points inside of a triangle. We cut this triangle into $k$ number of triangles so that

none of the triangles include a point

in every triangle's perimeter there are exactly three ponts: the three vertices

each triangle's vertices are from the $10$ points and the big triangle's vertices.

Find the maximum value of $k$ .

The answer is 21.

2 solutions

Áron Bán-Szabó
Jun 11, 2017

Suppose there are $k$ number of trangles. Then

$180°+360°(10)=180°(k)$

because there are $360°$ around each point and the sum of a triangle's angles is $180°$ .

Solving the equation we get $k=21$ .

This shows that if the solution exists, it should be $21$ triangles. It does not show that the solution does exist, or how to come up with the triangles.

Marta Reece - 3 years, 12 months ago

I think you mean that it is necessary for a newly formed triangle to have one of it's vertices shared by the big one as well.

Vishwash Kumar ΓΞΩ - 3 years, 12 months ago

Marta Reece
Jun 10, 2017

At the start there is one triangle.

Adding one point to the inside will result in three triangles, that is an addition of two.

Adding another point into any of these triangles will again produce three out of one, that is an addition of two.

Generally the number of triangles will continue to increase by $2$ for every point added.

The total number of triangles $N$ will be $2$ times the number of points $P$ , plus the one triangle that was there at the start.

$N=2\times P+1$

For $P=10$ this is $N_{10}=2\times 10+1=\boxed{21}$

Points in a triangle

The answer is 21.

2 solutions

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