Question

Prove that the intersection of any two perpendicular bisectors of
the sises of a triangle is the center of a circumscribing
circle.

Answer #1

Thanks

Need to Show that for any triangle, the angle bisectors
intersect. Then, show that the intersection point of the medians,
the intersection point of the altitudes, and the intersection point
of the angle bisectors lie on the same line.

Use any tools to construct an acute scalene triangle and the
three perpendicular bisectors, three angle bisectors, three
medians, and three altitudes. What do you notice?

Prove that the bisectors of a triangle are concurrent

Consider the triangle ABC. Suppose that the perpendicular
bisectors of line segments AB and BC intersect at point X. Prove
that X is on the perpendicular bisector of line segment AC.

Prove that the internal bisectors of the angles of a
triangle are concurrent.

Write up a formal proof that the perpendicular bisectors of a
triangle are concurrent, and that the point of concurrency (the
circumcenter) is equidistant from all three vertices

Draw a circle with your compass. Draw in a diameter. Construct a
perpendicular bisector of the diameter. Construct angle bisectors
of all four angles at the center of the circle. Mark where all
these lines intersect the circle. Connect these points. Prove that
the shape you have constructed is a regular octagon.

Are the perpendicular bisectors of the sides of a cyclic
quadrilateral concurrent? Sketch an example and make a conjecture.
What would you need to know to prove your conjecture?

Perpbisogram Definition: A perpbisogram of a
quadrilateral is a polygon formed by the perpendicular bisectors of
the sides of the quadrilateral.
Prove the following theorem
1) The perpbisogram of a rhombus is a rhombus.
SHOW
DIAGRAM

Prove that perpendicular drop to a chord from the center of a
circle bisects the chord.

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