Question

point a is the incenter of triangle def. which must be true? select three options
point a is the center of the circle that passes through points d, e, and f.
point a is the center of the circle that passes through points x, y, and z.
overline{za}congoverline{ya}
overline{ea}congoverline{fa}
overline{ax}congoverline{ay}

Explanation:

1. Recall the definition of the incenter: The incenter of a triangle is the intersection of the angle bisectors and is equidistant from all sides of the triangle. The circle centered at the incenter and tangent to the sides is the incircle, with the points of tangency being \(X\), \(Y\), \(Z\) (since \(AX\perp DE\), \(AY\perp DF\), \(AZ\perp EF\)).
2. Analyze each option:

• Option 1: The center of the circle passing through \(D\), \(E\), \(F\) is the circumcenter, not the incenter. So this is false.
• Option 2: The incenter is the center of the incircle, which is tangent to the sides at \(X\), \(Y\), \(Z\). So the circle centered at \(A\) passing through \(X\), \(Y\), \(Z\) (the incircle) is correct. This is true.
• Option 3: \(ZA\) and \(YA\) are both distances from the incenter to the sides \(EF\) and \(DF\) respectively. Since the incenter is equidistant from all sides, \(ZA = YA\), so \(\overline{ZA}\cong\overline{YA}\). This is true.
• Option 4: \(EA\) and \(FA\) are angle bisectors, but there's no reason they should be congruent (unless the triangle is isoceles in a specific way, which isn't given). So this is false.
• Option 5: \(AX\) and \(AY\) are both radii of the incircle (distances from incenter to the sides \(DE\) and \(DF\) respectively). Since the incenter is equidistant from all sides, \(AX = AY\), so \(\overline{AX}\cong\overline{AY}\). This is true.

Answer:

B. Point A is the center of the circle that passes through points X, Y, and Z.
C. \(\overline{ZA}\cong\overline{YA}\)
E. \(\overline{AX}\cong\overline{AY}\)