## Lecture Notes 4

Thm 4.1 : The perpendicular bisectors of the sides of a triangle are concurrent at a point called the circumcenter.

Note that in the pedal triangle the perp. bisectors are altitudes. This gives proof of

Thm 4.2 : The altitudes of a triangle are concurrent at a point called the orthocenter.

Thm 4.3 : The internal bisectors of the angles of a triangle meet at a point called the incenter.

Thm 4.4 : The medians of a triangle meet at a point called the centroid.

Thm 4.5 : The internal bisector of an angle of a triangle divides the opposite side into two segments proportional to the sides of the triangle adjacent to the angle.

Thm 4.6 : The external bisectors of two angles of a triangle meet the internal bisector of the third angle at a point called the excenter.

Thm 4.9 : The product of the lengths of the segments from an exterior point to the points of intersection of a secant with a circle is equal to the square of the length of a tangent to the circle from that point.

Heron's formula for the area of a triangle.

Note Brahmagupta's formula for the area of a cyclic quadrilateral.

Note the convention on directed line segments. Internal ratios are positive and external ones are negative.

Thm 4.10 : Menelaus's Theorem. If three points, one on each side of a triangle are collinear, then the product of the ratios of the division of the sides by the points is -1.

Pf. Drop perps from vertices to line and consider similar triangles.

Thm 4.11 : Converse of Menelaus's Theorem.

Look at point determined by two of the points and a side of triangle ...show it is the third point.

Thm 4.13 : Ceva's Theorem. Three lines joining vertices to points on the opposite sides of a triangle are concurrent if and only if the product of the ratios of the division of the sides is 1.

Thm 4.14 : The lines from the points of tangency of the incircle to the vertices of a triangle are concurrent (Gergonne point).

Thm 4.15 : The three perpendiculars from a point on the circumcircle to the sides of a triangle meet those sides in collinear points. The line is called the Simson line. (No proof)

Thm 4.19 : If three points are chosen, one on each side of a triangle then the three circles determined by a vertex and the two points on the adjacent sides meet at a point called the Miquel point.

Thm 4.16 : The midpoints of the sides of a triangle, the feet of the altitudes and the midpoints of the segments joining the orthocenter and the vertices all lie on a circle called the nine-point circle
For a proof consider the 9-point circle worksheet.

Thm 4.17: The centroid of a triangle trisects the segment joining the circumcenter and the orthocenter. [Euler line]

Thm 4.18: The center of the nine-point circle bisects the segment of the Euler line joining the orthocenter and the circumcenter of a triangle.

Thm 4.26: (Morley's Theorem) The adjacent trisectors of the angles of a triangle are concurrent by pairs at the vertices of an equilateral triangle.