## Math Club Geometry Training Session

Posted in Uncategorized by pgadey on 2019/01/31

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:

• 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 1 Congurence conditions for triangles: (SAS, ASA, SSS, AAS)
Side-angle-side, angle-side-angle, side-side-side, angle-angle-side.

Question 2 Why is angle-side-side not a congruence condition?

Fact 3 If an angle ${\angle A}$ measures a straight line, then ${\angle A = \pi = 180^\circ}$.

Question 4 Show that ${\angle ECD = \angle ACB}$ using the Fact~3.

Fact 5 (Transversals) ${AB}$ and ${CD}$ are parallel iff ${\angle B = \angle D}$.

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

Question 7 For any triangle ${\triangle ABC}$, the exterior angle at ${A}$ is the sum of the interior angles at ${B}$ and ${C}$.

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 ${\pi}$).
• The diagonals of a parallelogram bisect each other.
(Idea: Introduce triangles.)

Question 9 (Isoceles triangles) A triangle ${\triangle ABC}$ is isoceles if ${|AB| = |AC|}$. Show that: If ${\triangle ABC}$ is isoceles then ${\angle B = \angle C = 90^\circ - (\angle A)/2}$.

Question 10 Let ${\triangle ABC}$ be a triangle. Let ${D}$, ${E}$, and ${F}$ be the midpoints of ${BC}$, ${AC}$, and ${AB}$ respectively. Show that the lines: ${DE}$, ${EF}$, and ${FD}$, dissect ${\triangle ABC}$ in to congruent triangles.

(Big Idea: Introduce parallels to ${BC}$ and ${AC}$ at ${D}$. These will create new “phantom points” on ${BC}$ and ${AC}$. 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 13 Suppose that ${A}$ is outside a circle. If ${AX}$ and ${AY}$ are tangent to the circle, then ${|AX|=|AY|}$.

Question 14 (Inscribed Angle Theorem) Consider a circle centered at ${O}$ with points ${A,B,C}$ on the perimeter of the circle. Show that ${\angle ABC = 2\angle AOC}$. (Idea: Chase angles.)

Question 15 Consider a semi-circle with base ${AC}$.
If ${B}$ is on the perimeter of the semi-circle, show that ${\angle ABC = 90^\circ}$.

Question 16 (Circumcircles) The circumcircle of a triangle ${\triangle ABC}$ is a circle passing through ${A}$, ${B}$, and ${C}$.

• 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 ${\triangle ABC}$ is a circle tangent to ${AB}$, ${BC}$, and ${AC}$.

• 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 ${C}$ be a circle and ${P}$ a point in the plain. Each line through ${P}$ that interesects ${C}$ determines a chord of ${C}$. Show that the midpoints of these chords form a circle.

Question 19 (Rochester 2012) Let ${\triangle ABC}$ be an isoceles triangle with ${AC = BC}$ and ${\angle ACB = 80^\circ}$. Consider and interior point of this triangle such that ${\angle MBA = 30^\circ}$ and ${\angle MAB = 10^\circ}$. Find with proof, the measure ${\angle AMC}$.

Question 20 (Rochester 2013) Let ${\triangle ABC}$ be a triangle with ${\angle BAC = 120^\circ}$. Suppose the bisectors of ${\angle BAC}$, ${\angle ABC}$, and ${\angle ACB}$ meet ${BC}$, ${AC}$, and ${AB}$ at points ${D}$, ${E}$, ${F}$, respectively. Prove that ${\triangle DEF}$ is a right angled triangle.