Point A is the incenter of triangle DEF. Point A is the incenter of triangle D E F. Lines are drawn from the points of the triangle to point A. Lines are drawn from point A to the sides of the triangle to form right angles and line segments A X, A Y, and A Z. Which must be true? Select three options

The points must be true for Point A which is the incenter of triangle DEF are 1> Point A is the center of the circle that passes through points X, Y, and Z. 2>Line segment A X is-congruent-to line segment A Y . 3> Point A is always inside triangle DEF .

What is incenter of triangle?

The incenter of a triangle is the point of intersection of all the three interior angle bisectors of the triangle. This point is equidistant from the sides of a triangle, as the central axis's junction point is the center point of the triangle's inscribed circle. The incenter of a triangle is also known as the center of a triangle's circle since the largest circle could fit inside a triangle. The circle that is inscribed in a triangle is called an incircle of a triangle.

According to the question

Point A is the incenter of triangle DEF .

Lines are drawn from point A to the sides of the triangle to form right angles and line segments A X, A Y, and A Z. (X,Y,Z are points on middle of sides )

Now,

According to the properties of incenter of the triangle

1.> The circle that is inscribed in a triangle is called an incircle of a triangle.

i.e

Point A is the center of the circle that passes through points X, Y, and Z.

2> The incenter of a triangle is equidistant from the sides of a triangle, as the central axis's junction point is the center point of the triangle's inscribed circle.

i.e

Line segment A X is-congruent-to line segment A Y .

3> A triangle's incenter always lies inside the triangle.

i.e

Point A is always inside triangle DEF .

Hence, The points must be true for Point A which is the incenter of triangle DEF are 1> Point A is the center of the circle that passes through points X, Y, and Z. 2>Line segment A X is-congruent-to line segment A Y . 3> Point A is always inside triangle DEF .

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