## Math Club Geometry Training Session

These are some questions that I prepared for Math Club. The problems follow Paul Zeitz’s excellent book The Art and Craft of Problem Solving. You can find this hand-out is here:

https://pgadey.ca/teaching/2019-math-club/geometry-training-talk/geometry-training-talk.pdf (tex)

**1. Advice and Suggestions **

- Draw big diagrams. Lots of them.
- Use multiple colours to keep track on information.
- Draw several examples with different lengths/angles.
- Avoid coordinates.

**2. Facts and Questions **

Fact 1Congurence conditions for triangles: (SAS, ASA, SSS, AAS)

Side-angle-side, angle-side-angle, side-side-side, angle-angle-side.

Question 2Why is angle-side-sidenota congruence condition?

Question 4Show that using the Fact~3.

Fact 5 (Transversals)and are parallel iff .

Question 6Show that the sum of the interior angles of a triangle is . (Idea: Add a new parallel line.)

Question 7For any triangle , the exterior angle at is the sum of the interior angles at and .

Question 8 (Parallelograms)Use congruence and parallels to show:

- The opposite sides of a parallelogram have the same length.
- The opposite angles of a parallelogram are equal, and adjacent angles are “supplementary” (sum to ).
- The diagonals of a parallelogram bisect each other.

(Idea: Introduce triangles.)

Question 9 (Isoceles triangles)A triangle is isoceles if . Show that: If is isoceles then .

Question 10Let be a triangle. Let , , and be the midpoints of , , and respectively. Show that the lines: , , and , dissect in to congruent triangles.

(Big Idea: Introduce parallels to and at . These will create new “phantom points” on and . These new points will be very helpful because of our theory of parallelograms.)

Fact 11 (Circles and Chords)If any two are true, then all three are true.

- The line passes through the center of the circle.
- The line passes through the midpoint of the chord.
- The line line is perpendicular to the chord.

Fact 12 (Circles and Tangents)A tangent to a circle is perpendicular to the radius at the point of tangency. Also, a perpendicular to a tangent line placed at the point of tangency, will pass through the center of the center of the circle.

Question 13Suppose that is outside a circle. If and are tangent to the circle, then .

Question 14 (Inscribed Angle Theorem)Consider a circle centered at with points on the perimeter of the circle. Show that . (Idea: Chase angles.)

Question 15Consider a semi-circle with base .

If is on the perimeter of the semi-circle, show that .

Question 16 (Circumcircles)The circumcircle of a triangle is a circle passing through , , and .

- Suppose that a circumcircle exists. Show that its center is the intersection point of the perpendicular bisectors of the sides of the triangle. (If a circumcircle exists, then it has a unique center.)
- Given a triangle, any two perpendicular bisectors will intersect in a point equidistant from all three vertices. (Any triangle has a circumcenter.)
- All three perpendicular bisectors intersect in a unique point, the “circumcenter”.

Question 17 (Incircles)The incircle of a triangle is a circle tangent to , , and .

- Suppose that an incircle exists. Show that its center is the intersection of any two angle bisectors.
- Show that the intersection of any two angle bisectors is the center of an incircle and this point is unique.
- All three angle bisectors intersect in a unique point, the “incenter”.

**3. Contest Problems **

Question 18 (Canada 1991)Let be a circle and a point in the plain. Each line through that interesects determines a chord of . Show that the midpoints of these chords form a circle.

Question 19 (Rochester 2012)Let be an isoceles triangle with and . Consider and interior point of this triangle such that and . Find with proof, the measure .

Question 20 (Rochester 2013)Let be a triangle with . Suppose the bisectors of , , and meet , , and at points , , , respectively. Prove that is a right angled triangle.

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