2. Nine Point Circle.
Given a triangle ABC. Construct the orthocenter and locate poins R, S, and T as the midpoints of the segments from the orthocenter to each vertex. Construct a circle through R, S, and T. GSP file if you want it.
Prove the circle goes through K, the midpoint of BC.
Prove the circle goes through D the foot of the altitude from vertex A.
5. Areas formed by the Medians of a Triangle
7. Construct Triangle Given Its Medians
9. Simson Line
11. 20-30-150 Triangle Problem
12. Product of Chord parts:
Given a circle and two chords AB and CD intersecting at point P in the circle, then the products of the parts of each chord are equal. That is:
(AP)(PB) = (CP)(PD)
GSP File? Note that this product and theorem could be about the diagonals of an inscribed quadrilateral.
13. 100 Degree Isosceles Triangle
14. Tangent/Secant Product.
Given a circle and a tangent segment PT and secant AB from the same external point P, the square of the tangent segment is equal to the product of the secant segment PA and its external portion PB. That is:
15.
Alternative Proof of the Pythagorean Theorem.
Take any right triangle ABC with C at the right angle. Use the the Tangent/Secant product theorem (Problem 14) to prove the Pythagorean Theorem as suggested (perhaps) by the sketch at the right.
16. Secant/secant product theorem
Given a circle and two secants from an external point P, the product of the measures on one secant segment and its external portion is equal to the product of the measures of the other segment segment and its external portion. That is:
(PA)(PB) = (PC)(PD)
17. Ptolemy's Theorem
In a cyclic quadrilateral, the product of the diagonals is equal to the sum of the products of pairs of opposite sides. That is,
(BD)(AC) = (AD)(BC) + (AB)(CD)
Converse of Ptolemy's Theorem
Given a quadrilateral ABCD with the sum of the products of the two pairs of opposite sides equal to the product of the two diagonals. That is, AD·BC + AB·CD = AC·BD. Prove that ABCD is inscribed in a circle.
18. Ceva's Theorem
Given a triangle ABC with an arbitrary point P inside the triangle. Any segment from a vertex to the intersection with the opposite side is called a Cevian. Let D, E, and F be the feet of the Cevians opposite A, B,and C respectively. GSP file?
If the three Cevians AD, BE, and CF are concurrent at P, then prove
Converse. If
then the Cevians AD, BE, and CF are concurrent.
Most proofs of this theorem in textbooks and out on the web construct auxiliary lines to develop similar triangles. A far simpler proof can be developed from the fact that areas with the same altitude are proportional to the lengths of the bases. You are encouraged to find an area proof.
19. Proportional Adjacent sides to a vertex
Theorem: Given a set of triangles all having the same base AB. What is the locus of the vertex C in the ratio of the sides adjacent to C is 1? Proof?Theorem: Given a set of triangles all having the same base AB. What is the locus of the vertex C in the ratio of the sides adjacent to C is not equal to 1? Proof? Build a GSP animation showing this locus.
20. Law of Sines
