Download propositions on circles and triangles

Math 460
Euclidean Circles and Triangles
Law of Cosines. In any triangle ABC, with sides a, b, and c and angle θ at C: c2 = a2 + b2 - 2ab cos θ
Law of Sines. In any triangle ABC, with sides a, b, and c and angles α, β, and γ at A, B and C:
sin a sin b sin g
=
=
a
b
c
Prop 2.1. In a right triangle ABC, with hypotenuse AB let M be the midpoint AB. Then
AM = BM = CM.
Prop 2.2. In any triangle ABC, let D and E be the midpoints of AB and AC respectively. Then DE is
parallel to BC and 2DE = BC.
Prop 2.3. Any three non-collinear points determine a circle.
Definition A tangent line to a circle cuts it at one point.
Prop 2.4. If a tangent line cuts a circle (center C) at T, then the tangent line is orthogonal to CT.
Prop 2.5. Let A and B be on a circle centered at C. Then the arc AB subtends the same argle for any
point D on the circle. This angle is ½ the central angle ‫ ﮮ‬ACB
Corollary: In a circle with diameter AB, and P any other point on the circle, the triangle ABP has a
right angle at P.
Prop 2.6. Let C be the locus of all points P that subtend a right angle on AB. Then C is a circle with
diameter AB.
Prop 2.7. Given a circle centered at C and a point P outside the circle, a tangent line from P to the
circle can be constructed.
Definition. The altitudes of a triangle are the lines through the vertices which are perpendicular to the
opposite sides. The foot of the altitude is the the intersection of the altitude with the (possibly extended)
opposite side.
Definition. In any triangle ABC, you can determine four points called the incenter, the orthocenter,
the centroid and the circumcenter.
The three angle bisectors are concurrent at the incenter.
The three medians are concurrent at the centroid.
The three perpendicular bisectors are concurrent at the circumcenter
The three altitudes concurrent at the orthocenter
Prop 2.8. In any triangle ABC, let D be the intersection point of 2 of the angle bisectors from A and
B. Then the three perpendiculars from D to each of the three sides of the triangle are all equal in length.
Then the angle bisector from C also contains D and the circle centered at D is tangent to each the sides
of ABC.
Prop 2.9. In any triangle ABC, let the medians be AD, BE and CF. Then they intersect at a point G and
AG = 2GD, BG = 2GE and CG = 2GF. G is the centroid of the triangle.
Prop 2.10. In any triangle ABC, let the midpoints of the sides be D, E F. Then the three perpendicular
bisectors of the sides intersect at a single point O (the circumcenter), which is the center of the circle
containing points A, B and C. (This is essentially a restatement of prop 2.3.
Prop 2.11. In any triangle ABC, the altitudes from A, B and C all intersect at a single point H, the
orthcenter.
Prop 2.12. In any triangle ABC, the circumcenter, the centroid and the orthocenter are collinear. This
line is known as the Euler line.
Prop 2.13. In any triangle ABC, let N be the midpoint of the orthocenter and the circuncenter. N lies
on the Euler line and is the center of the nine point circle. This circle intersects the three midpoints of
the sides, the three midpoints of the vertices with the orthocenter and the three feet of the altitudes.