Plane Geometry

Peter McNamara December 2004

If you look in your copy of the Toolchest, there is a geometry section which contains most of the theory that you should know. What I have included here is not complete, just a sample of some of what I think are important facts you should know plus a couple of results that are not in the toolchest.

0.1 Theorems

1. Two triangles are congruent if

(a) SSS Rule: all three sides are equal

(b) SAS Rule: two sides and the included angle are equal

(c) ASA Rule: two angles and the included side are equal

(d) RHS Rule: the hypotenuse and one leg of a right triangle are equal

2. Two triangles are similar if

(a) (AA) two angles (and thus all three) are equal

(b) (SSS) all three sides are in proportion

(c) (SAS) two sides are in proportion and the included angle is equal

3. The sum of the interior angles of an n-gon is 180(n − 2) degrees.

4. If four points A, B, C and D are concyclic, then ∠ACB = ∠ADB and ∠ABC + ∠CDA = 180. (This can also be used as a criterion for cyclicity)

5. (alternate segment theorem) If A and B are points on the circumference of a circle with centre O and C is such that BC is tangent to this circle, then ∠ABC = \frac{1}{2}∠AOB (and conversely).

6. (power of a point) Let Γ be a circle with centre O, radius R and let P be a point in the plane. If two lines through P intersects Γ in A and B and C and D respectively, then P A · P B = P C · P D = |R^{2} − OP^{2}|. If a tangent P T can be drawn from P to Γ with T on Γ, this quantity is also equal to P T^{2}. (Again, this can be used as a criterion for concyclicity)

7. (radical axis theorem) The radical axis of two circles is defined to be the set of points which has equal power (as in the power of a point) with respect to both circles. The locus of such points is a line. The radical axis theorem states that given three circles in the plane, they determine three radical axes and these radical axes are concurrent.

8. (angle bisector theorem) In a triangle ABC, if the interior and exterior angle bisectors of angle A meet the line BC at D and E respectively, then

\frac{BD}{CD} = \frac{AB}{AC} = \frac{BE}{CE}

9. (Ceva’s theorem) Let D, E and F lie on the sides (possibly extended) BC, CA and AB respectively of △ABC. Then AD, BE and CF are concurrent if and only if

\frac{BD}{DC} · \frac{CE}{EA} · \frac{AF}{F B} = 1

10. (Menelaus’ theorem) Let D, E and F lie on the sides (possibly extended) BC, CA and AB respectively of △ABC. Then D, E and F are collinear if and only if

\frac{BD}{DC} · \frac{CE}{EA} · \frac{AF}{F B} = −1

11. (Ptolemy’s Inequality) In a quadrilateral ABCD, AC · BD ≤ AB · CD + BC · DA with equality if and only if the quadrilateral is cyclic.

12. (Brahmagupta) In a quadrilateral with side lengths a, b, c, d, semiperimeter s = (a + b + c + d)/2 and area Δ, we have Δ ≤ \sqrt{(s − a)(s − b)(s − c)(s − d)} with equality if and only if the quadrilateral is cyclic.

0.2 Problems

1. Let O be the circumcentre of △ABC. Prove that ∠AOB = 2∠ACB.

2. D and E are two points on the side BC of triangle ABC such that ∠BAD = ∠EAC and |AD| = |AE|. Prove that ABC is isosceles.

3. Prove that AC is perpendicular to BD if and only if AB^{2}+CD^{2} = BC^{2}+DA^{2}.

4. Let D be the foot of the altitude from A in triangle ABC. Prove that △ABC is right-angled at A if and only if AD^{2} = BD · DC.

5. Prove that in a triangle ABC with orthocentre H, △ABH has the same circumradius as △ABC.

6. (Tournament of The Towns, Junior O level, Autumn 1997) Let ABCD be a square and X, Y be points on the sides BC and CD respectively such that AY bisects ∠DAX. Show that AX = BX + DY.

7. (1998 AMOC Senior Contest) Triangle Δ has side lengths a, b, c, triangle Δ_{a} has side lengths a′, b, c, triangle Δ_{b} has side lengths a, b′, c, triangle Δ_{c} has side lengths a, b, c′, and each of the four triangles has area 1. Furthermore a ≠ a′, b ≠ b′, c ≠ c′.

(a) Prove that there exists a triangle, Δ′ say, with side lengths a′, b′, c′.

(b) Determine the area of triangle Δ′.

8. (1999 AMOC Senior Contest) Circle k_{1} has its centre on another circle, k_{2}. The circles intersect at A and C. From any point B on k_{2}, draw BC, intersecting k_{1} again at D. Prove that AB = BD.

9. Let A, B, C, D be concyclic points in that order. Show that

\frac{AB · BC + AD · DC}{AB · AD + BC · CD} = \frac{BD}{AC}

10. Prove that a quadrilateral with side lengths a, b, c and d in that order can have a circle inscribed inside it tangent to all four sides if and only if a + c = b + d.

11. Prove Ceva’s Theorem

12. (1998 AMO Q3) Let ABC be a triangle, let D be a point on AB and E be a point on AC such that DE and BC are parallel and DE is a tangent to the incircle of the triangle ABC. Prove that 8DE ≤ AB + BC + CA.

13. (1999 AMO Q6) Let ABC be a triangle and D, E, F points in its exterior such that △ABD, △BCE and △CAF are equilateral. The sides of these triangles are extended to produce the following intersections: BE and AF intersect in K, DB and F C intersect in L, and DA and EC intersect in M. Prove that DK, EL and F M are parallel.

14. Let ABC be a triangle and l_{a}, l_{b} and l_{c} be three parallel lines through A, B and C respectively. Let m_{a}, m_{b} and m_{c} be the reflections of l_{a}, l_{b} and l_{c} about the internal angle bisectors at A, B and C respectively. Prove that m_{a}, m_{b} and m_{c} are concurrent and find the locus of all points of concurrency.

15. The points P and Q lie on a semicircle with diameter UV. UP and V Q intersect at the point S, while the tangents to the semicircle at P and Q intersect at R. Prove that RS ⊥ UV.

16. (1991 IMO Q1) Given a triangle ABC, let I be the centre of its inscribed circle. The internal bisectors of the angle A, B, C meet the opposite sides in A′, B′, C′ respectively. Prove that \frac{1}{4} < \frac{AI·BI·CI}{AA′·BB′·CC′} ≤ \frac{8}{27}.

17. (1997 IMO Q2) Angle A is the smallest in the triangle ABC. The points B and C divide the circumcircle of the triangle into two arcs. Let U be an interior point of the arc between B and C that does not contain A. The perpendicular bisectors of AB and AC meet the line AU at V and W respectively. The lines BV and CW meet at T. Show that AU = T B + T C.

18. (2002 IMO Q2) Let BC be a diameter of a circle Γ with centre O. Let A be a point on Γ such that 0 < ∠AOB < 120^{∘}. Let D be the midpoint of the arc AB not containing C. The line through O parallel to DA meets the line AC at J. The perpendicular bisector of OA meets Γ at E and F. Prove that J is the incentre of the triangle CEF.