Parker Glynn-Adey

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:
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 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.

vertical-angles

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

transversal

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.)

inscribed-angle-theorem

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.

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