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BERKELEY MATH CIRCLE 1999-2000

Classical Theorems in Plane Geometry¹

ZVEZDELINA STANKOVA-FRENKEL
UC BERKELEY AND MILLS COLLEGE

Note: All objects in this handout are planar - i.e. they lie in the usual plane. We say that several points are collinear if they lie on a line. Similarly, several points are concyclic if they lie on a circle; an inscribed (cyclic) polygon has its vertices lying on a circle. If three distinct points A, B and C are collinear, then the directed ratio $\overline{AB}/\overline{CB}$ is the ratio of the lengths of segments AB and CB, taken with a sign ``+'' if the segments have the same direction (i.e. B is not between A and C), and with a sign ``-'' if the segments have opposite directions (i.e. B is between A and C). Several objects (lines, circles, etc.) are concurrent if they all intersect in some point.

1.

(Menelaus) Let A₁,B₁ and C₁ be three points on the sides BC,CA and AB of $\triangle ABC$. Prove that they are collinear (cf. Fig. 1) iff

$$\frac{\overline{AB_1}}{\overline{CB_1}} \cdot\frac{\overline{... ...overline{BA_1}} \cdot\frac{\overline{BC_1}}{\overline{AC_1}}=1.$$

2.

(a)

Prove that the interior angle bisectors of two angles of nonisosceles $\triangle ABC$ and the exterior angle bisector of the third angle intersect the opposite sides (or their continuations) of $\triangle ABC$ in three collinear points. (cf. Fig. 2a)

(b)

Prove that the exterior angle bisectors of nonisosceles $\triangle ABC$ intersect the continuations of opposite sides of $\triangle ABC$ in three collinear points.

(c)

Prove that the tangents at the vertices of nonequilateral $\triangle ABC$ to the circumcircle of $\triangle ABC$ intersect the continuations of opposite sides of $\triangle ABC$ in three collinear points. (cf. Fig. 2c)

3.

(Pascal) If the hexagon ABCDEF is cyclic and its opposite sides, AB and DE, BC and EF, CD and FA, are pairwise not parallel, prove that their three points of intersection, X, Y and Z, are collinear. (cf. Fig. 3)

The same statement is true if the circle is replaced by an ellipse, hyperbola or parabola.² The statement is also true if some of the vertices of the hexagon coincide - then replace the corresponding side of the hexagon by the tangent to the circle at the corresponding vertex. Thus, obtain the following:

(a)

If A=B, C=D, D=F, deduce to Problem 2c.

(b)

If E=F, formulate the property of any inscribed pentagon.

(c)

If A=F and D=E, for the inscribed quadrilateral ABCDwe have: the intersection points of AB and the tangent at D, of CD and the tangent at A, and of BC and AD, are collinear.

(d)

If A=F and C=D, the intersection points of the pairs of opposite sides of an inscribed quadrilateral and the intersection of the tangents at two opposite vertices are collinear. (Actually, the tangents at any pair of opposite vertices should also work.)

4.

(Desargues) $\triangle ABC$ and $\triangle A_1B_1C_1$ are positioned in such a way that lines AA₁, BB₁, and CC₁ intersect in a point O. If lines AB and A₁B₁, AC and A₁C₁, BC and B₁C₁ are pairwise not parallel, prove that their points of intersection, L, M and N, are collinear. (cf. Fig. 4)

Note: The incircle of $\triangle ABC$ is the circle inscribed in $\triangle ABC$ (i.e. tangent to all three sides of the triangle.) Its center is called the incenter of $\triangle ABC$; it lies on the angle bisectors of $\triangle ABC$. The excircle of $\triangle ABC$ tangent to side AB is the circle tangent to side AB and to the extentions of sides BCand AC. Its center is called an excenter of $\triangle ABC$. On what bisectors does this excenter lie?

Finally, the circumcircle of $\triangle ABC$ is the circle passing through the vertices A, B and C. Its center is called the circumcenter of $\triangle ABC$; it lies on the perpendicular bisectors of the sides of the triangle.

5.

Prove that the midpoint K of the altitude CH in $\triangle ABC$, the incenter I of $\triangle ABC$, and the tangency point T on AB of the excircle of $\triangle ABC$ (tangent to side AB) are collinear. (cf. Fig. 5)

6.

(Gauss's line with respect to l) Line lintersects the sides (or continuations of) BC, CA and AB of $\triangle ABC$ in points P₁, P₂ and P₃. Prove that the midpoints M₁, M₂ and M₃ of AP₁, BP₂ and CP₃ are collinear. (cf. Fig. 6)

7.

Let ABCD be a quadrilateral with perpendicular diagonals intersecting in P. The feet of the perpendiculars from Pto sides AB, BC, CD and DA are P₁, P₂, P₃ and P₄. Prove that lines P₁P₂, P₃P₄ and CA are concurrent. (cf. Fig. 7)

8.

(Simpson) Prove that the feet of the perpendiculars dropped from a point M on the circumcircle k of $\triangle ABC$ to the sides of the triangle are collinear. More generally, let S be the area of $\triangle ABC$, R - the circumradius, and d - the radius of a circle $\epsilon$ concentric to k. Let A₁, B₁ and C₁ be the feet of the perpendiculars dropped from an arbitrary point on $\epsilon$ to the sides of $\triangle ABC$. Prove that the area S₁ of $\triangle A_1B_1C_1$ is given by the formula $S_1=\frac{1}{4}S\big\vert 1-\frac{d^2}{R^2}\big\vert.$ In particular, when $\epsilon=k$, then S₁=0, and hence A₁, B₁ and C₁ are collinear. (cf. Fig. 8)

9.

(Salmon) Through a point M on a circle $\epsilon$ draw three arbitrary chords MA, MB and MC, and using each chord as a diameter, draw three new circles $\epsilon_1$, $\epsilon_2$, and $\epsilon_3$. Prove that the pairwise intersections of the $\epsilon_i$'s (other than M) are collinear.

Note: Let H be the orthocenter of $\triangle ABC$(i.e. the intersection of the altitudes of the triangle.) The Euler circle of 9 points for $\triangle ABC$ is the circle passing through the midpoints of the sides of $\triangle ABC$, the midpoints of AH, BH and CH, and the feet of the altitudes of $\triangle ABC$. In fact, the center of this circle is the midpoint of HO (Ois the circumcenter of $\triangle ABC$), and its radius is half of the circumradius of $\triangle ABC$. Why? (cf. Fig. 10a)

10.

Prove that Simpson's line of $\triangle ABC$ with respect to point M on the circumscribed circle k of $\triangle ABC$, line MH where H is the orthocenter of $\triangle ABC$, and the Euler circle of 9 points for $\triangle ABC$ are concurrent. (cf. Fig. 10b)

11.

(Ceva) Let $A^{\prime}$, $B^{\prime}$ and $C^{\prime}$ be three points on the sides (or continuations of) BC, CA, AB of $\triangle ABC$. Prove that $AA^{\prime}, BB^{\prime}, CC^{\prime}$ are concurrent or are parallel iff

$$\frac{\overline{AB^{\prime}}}{\overline{CB^{\prime}}} \cdot\f... ... \cdot\frac{\overline{BC^{\prime}}}{\overline{AC^{\prime}}}=-1.$$

12.

(Gergonne's point) Prove that the lines connecting the vertices of a triangle with the points of tangency of the inscribed circle are concurrent. (cf. Fig. 12)

13.

(Nagel's point) Prove that the lines connecting the vertices of a triangle with the corresponding points of tangency of the three externally inscribed circles are concurrent (cf. Fig. 13.) Note that these are also the three lines through the vertices of the triangle and dividing each its perimeter into two equal parts.

14.

Let M be an arbitrary point on side AB of $\triangle ABC$. Let P and Q be the intersection points of the angle bisectors of $\angle BMC$ and $\angle AMC$ with sides BC and AC, respectively. Prove that lines AP, BQ and CM are concurrent.

15.

Let A₁,B₁,C₁ be points on the sides of an acuteangled $\triangle ABC$ so that the lines AA₁,BB₁and CC₁ are concurrent. Prove that CC₁ is an altitude in $\triangle ABC$ iff it is the angle bisector of $\angle B_1C_1A_1$.

16.

In the acuteangled $\triangle ABC$ a semicircle kwith center O on side AB is inscribed. Let M and N be the points of tangency of k with sides BC and AC. Prove that lines AM, BN and the altitude CD of $\triangle ABC$ are concurrent. (cf. Fig. 16)

17.

A circle k intersects side AB of $\triangle ABC$ in C₁ and C₂, side CA - in B₁ and B₂, side BC - in A₁ and A₂. The order of these points on k is: A₁, A₂, B₁, B₂, C₂, C₁. Prove that lines AA₁, BB₁, CC₁are concurrent iff AA₂, BB₂, CC₂ are concurrent. (cf. Fig. 17)

18.

Let the points of tangency of the incircle of $\triangle ABC$ with the sides AB, BC and CA be C₁, A₁ and B₁, and let A₂, B₂ and C₂ be their reflections across the incenter I of $\triangle ABC$. Prove that lines AA₂, BB₂ and CC₂ are concurrent. (cf. Fig. 18)

19.

(Gauss) If the two pairs of opposite sides of a quadrilateral ABCD intersect in E and F, prove that the midpoint N of EF lies on the line through the midpoints L and M of the diagonals AC and BD. (cf. Fig. 19)

20.

Point P lies inside $\triangle ABC$. Lines AP, BP, CP intersect the sides BC, CA, AB in A₁,B₁,C₁, respectively, and L,M,N,L₁,M₁,N₁ are the midpoints of the segments BC,CA,AB,B₁C₁,C₁A₁, A₁B₁. Prove that LL₁,MM₁ and NN₁ are concurrent. (cf. Fig. 20)

21.

Let P, Q and R be points on the sides BC, CA and AB of $\triangle ABC$. Let O₁, O₂ and O₃ be the circumcenters of $\triangle AQR$, $\triangle BRP$ and $\triangle CPQ$. Prove that $\triangle O_1O_2O_3\sim \triangle ABC$. (cf. Fig. 21)

22.

(IMO'81) Three congruent circles pass through point P inside $\triangle ABC$. Each circle is inside $\triangle ABC$ and is tangent to two of its sides. Prove that the circumcenter O and incenter I of $\triangle ABC$ and P are collinear. (cf. Fig. 22)

23.

(Brianchon) If the hexagon ABCDEF is circumscribed around a circle, prove that its three diagonals AD,BEand CF are concurrent. (cf. Fig. 23)

24.

(Saint Petersburg Olympiad) Point I is the incenter of $\triangle ABC$. Some circle with center I intersects side BC in A₁ and A₂, side CA in B₁ and B₂, and side AB in C₁ and C₂. The six points obtained in this way lie on the circle in the following order: $A_1,\,\,A_2, \,\,B_1,\,\,B_2,\,\,C_1,\,\,C_2$. Points $A_3,\,\,B_3$ and C₃ are the midpoints of the arc $A_1A_2,\,\,B_1B_2$ and C₁C₂respectively. Lines A₂A₃ and B₁B₃ intersect in C₄, lines B₂B₃ and C₁C₃ - in A₄, and lines C₂C₃ and A₁A₃ - in B₄. Prove that the segments $A_3A_4,\,\,B_3B_4$ and C₃C₄intersect in one point. (cf. Fig. 24)