Geometry: Triangles, Circles, and Polygons

The radius drawn to a point of tangency is perpendicular to the tangent line. Since the tangent is perpendicular to radius at the midpoint, and AB is a line segment, angle OAB is 90°, making triangle OAB a right triangle.

An equilateral triangle must have three 60-degree angles. A right triangle must have one 90-degree angle. Since a triangle cannot have both a 90-degree and three 60-degree angles, a right triangle can never be equilateral.

The bisection ray of an inscribed angle does not generally pass through the center of the circle unless the inscribed angle is isosceles. Therefore, the statement is false.

The bisection ray of an inscribed angle only passes through the center of the circle if the two chords forming the angle are equal in length. Since this is not true for all inscribed angles, the statement is false.

An angle inscribed in a circle is bisected by a ray. This ray only passes through the center of the circle if the inscribed angle is symmetric with respect to the center (i.e., its sides are chords of equal length). In general, it does not pass through the center, so the statement is false.

Connecting the non-adjacent trisection points of the sides of an isosceles triangle does not generally form an isosceles trapezoid. Thus, the statement is false.

In a rectangle, the diagonals are congruent and bisect each other, but they are only perpendicular if the rectangle is a square. Therefore, their perpendicular bisectors are not perpendicular in a general rectangle, making the statement false.

In a general triangle, the centroid (intersection of medians) and the incenter (intersection of angle bisectors) are different points. They only coincide in an equilateral triangle. Therefore, the statement is false.

For a regular polygon with n sides, the exterior angle is 360/n and the interior angle is 180 - 360/n. For a triangle (n=3), the exterior angle (120) is greater than the interior (60). For a square (n=4), they are equal. It is only less for n > 4. Thus, the statement is false in general.

Supplementary angles are defined as two angles whose measures sum to 180°, not 90°. Angles that sum to 90° are called complementary angles. Therefore, the statement 'Two angles are supplementary if they sum to 90 degrees' is false. The correct answer is False.

The orthocenter of a triangle is the intersection point of its three altitudes. In an obtuse triangle, the altitudes from the acute vertices lie outside the triangle, and their intersection point (the orthocenter) is indeed located outside the triangle. This is a well-known geometric property. The statement is true.

An equilateral triangle has three equal sides. If it shares two sides with a parallelogram, those two adjacent sides of the parallelogram must be equal to the sides of the triangle, and thus equal to each other. A parallelogram with equal adjacent sides is a rhombus. Thus, the statement is true.

Connecting the midpoints of an equilateral triangle creates a central triangle with 1/4 the area of the original. Repeating this process 3 more times (4 times total) results in an area of (1/4)^4 = 1/256 of the original area. Thus, the statement is true.

In an isosceles triangle, the base is the side of unequal length, while the two legs are the equal sides. By the triangle inequality theorem, any side of a triangle must be less than the sum of the other two sides. Therefore, the base cannot be greater than the sum of the two legs - it must be less than their sum. The statement is false, and the correct answer is False.