The triangle has a point of concurrency at p. find the value of x that would make p the incenter of the triangle. x = find the value of x that would make p the circumcenter of the triangle. x =

The value of x in the given circle determines if the point P is the

circumcenter or the incenter of the triangle.

Reasons:

Incenter of a triangle;

The incenter of a triangle is the center of an inscribed circle of the triangle

Therefore, given that x makes P, the incenter of the triangle, we have;

(3·x + 3) = 24

Therefore;

Therefore;

The value of x that will make the P the incenter of the triangle is x = 7°

Circumcenter of the triangle;

The circumcenter of the triangle, is the center of the circle circumscribing

the triangle.

Therefore, the length of the segments from P, to the vertices are the length

of the radius of the circumscribing circle and are therefore equal.

Which gives;

5·x - 4 = 26

The value of x, that makes P, the circumcenter of the triangle, is x = 6°

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Part a)

we know that

The incenter is the point forming the origin of a circle inscribed inside the triangle

so

in this problem

The radius of the inscribed circle is equal to

so

solve for x

therefore

the answer Part a) is

Part b)

we know that

The circumcenter is the center of the the circle that passes through all three of the triangle's vertices

so

In this problem

The radius of the circle is equal to

solve for x

therefore

the answer Part b) is

the answer is 40.

proof: