Point X is the incenter of ΔABC.
Triangle A B C has point X as its incenter. Lines are drawn from the points of the triangle to point X. Lines are drawn from point X to the sides of the triangle to form right angles. Line segments X F, X G, and X E are formed.

If EX = 4z + 1, XF = 2z + 7, and mABC = 44°, find the following measures.

GX =

mABX = °

We know that the point at which the triangle's three interior angle bisectors converge is known as the incenter. It can be thought of as the intersection of the triangle's internal angle bisectors. Due to the junction point of the central axis being the center of the triangle's inscribed circle, this point will be equally spaced from each of the triangle's sides. The center of a triangle's inscribed circle, which is the biggest circle that can fit inside the triangle, is known as the incenter.

Given that EX = 4z + 1 and XF = 2z + 7.

Since X is the incentre of this triangle ΔABC, EX = XF = GX.

Now, 4z + 1 = 2z + 7

i.e. 4z - 2z = 7 - 1

i.e. 2z = 6

i.e. z = 6/2 = 3

Then GX = 2 * 3 + 7 = 6 + 7 = 13

Also given that ∠ABC = 44°.

Since X is the incentre of this triangle ΔABC, ∠ABX = ∠CBX

Let ∠ABX = ∠CBX = y

Now y + y = 44

i.e. 2y = 44

i.e y = 44/2 = 22

Then ∠ABX = 22°

Therefore, the length of GX is 13 and the value of ∠ABX is 22°.

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