CLASSICAL THEOREMS IN PLANE GEOMETRY

Note: All objects in this handout are planar - i.e. they lie in the usual plane. We say that several
points are col linear 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 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
. Prove that they are collinear (cf. Fig. 1) iff

2. (a) Prove that the interior angle bisectors of two angles of nonisosceles and
the exterior angle bisector of the third angle intersect the opposite sides (or
their continuations) of in three col linear points . (cf. Fig. 2a)
(b) Prove that the exterior angle bisectors of nonisosceles intersect the
continuations of opposite sides of in three collinear points.
(c) Prove that the tangents at the vertices of nonequilateral to the circumcircle
of intersect the continuations of opposite sides of 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 . 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 fol lowing :

(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 ABCD we 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) and are positi oned 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 is the circle inscribed in (i.e. tangent to all three sides of the
triangle.) Its center is called the incenter of ; it lies on the angle bisectors of . The
excircle of tangent to side AB is the circle tangent to side AB and to the extentions of sides
BC and AC. Its center is called an excenter of . On what bisectors does this excenter lie?
The circumcircle of is the circle passing through the vertices A, B and C. Its center is called
the circumcenter of ; it lies on the perpendicular bisectors of the sides of the triangle.

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

6. (Gauss’s line with respect to l) Line l intersects the sides (or continuations of) BC,
CA and AB of 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 P to 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 to the sides of the triangle are collinear. More generally,
let S be the area of , R – the circumradius, and d – the radius of a circle
concentric to k. Let A₁, B₁ and C₁ be the feet of the perpendiculars dropped from
an arbitrary point on to the sides of . Prove that the area
is given by the formula . In particular, when , then ,
and hence A₁, B₁ and C₁ are collinear. (cf. Fig. 8)

9. (Salmon) Through a point M on a circle draw three arbitrary chords MA, MB
and MC, and using each chord as a diameter, draw three new circles
Prove that the pairwise intersections of the (other than M) are collinear.

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

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

11. (Ceva) Let A‘, B’ and C‘ be three points on the sides (or continuations of) BC,
CA, AB of . Prove that AA’,BB‘,CC’ are concurrent or are parallel iff

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 con-
current (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 . Let P and Q be the intersec-
tion points of the angle bisectors of 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 so that the lines
AA₁,BB₁ and CC₁ are concurrent. Prove that CC₁ is an altitude in iff it
is the angle bisector of .

16. In the acuteangled a semicircle k with 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 are concurrent. (cf. Fig. 16)

17. A circle k intersects side AB of 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, CC₁ are concurrent iff AA₂, BB₂, CC₂ are concurrent.
(cf. Fig. 17)

18. Let the points of tangency of the incircle of 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 . 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 . 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 and
be the circumcenters of
. (cf. Fig. 21)

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

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

24. (Saint Petersburg Olympiad) Point I is the incenter of . 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: A₁, A₂, B₁, B₂, C₁, C₂. Points A₃, B₃ and C₃ are the midpoints of the
arc A₁A₂, B₁B₂ and C₁C₂ respectively. Lines A₂A₃ and B₁B₃ intersect in C₄, lines
B₂B₃ and C₁C₃ – in A₄, and lines C₂C₃ and A₁A₃ – in B₄. Prove that the segments
A₃A₄, B₃B₄ and C₃C₄ intersect in one point. (cf. Fig. 24)

25. (Bulgarian IMO Test ’08) In let AM (M ∈BC) be a median and let CC₁
(C₁ ∈AB) and BB₁ (B₁ ∈AC) be two altitudes. The line through A perpendicular
to AM intersects lines BB₁ and CC₁ in points E and F, respectively. Denote by
k the circumcircle of be circles tangent to both EF and to
the arc EF on K not containing M. If P and Q are the intersection points of k₁
and k₂, prove that points P, Q, and M lie on a line.

26. (IMO ’08, Spain) An acute-angled has orthocenter H. The circle passing
through H with center the midpoint of BC intersects the line BC at A₁ and A₂.
Similarly, the circle passing through H with center the midpoint of CA intersects the
line CA at B₁ and B₂, and the circle passing through H with center the midpoint
of AB intersects the line AB at C₁ and C₂. Show that A₁, A₂, B₁, B₂, C₁, and C₂
lie on a circle.

27. (BAMO ’06) Given , let A₁, B₁, and C₁ be points on sides BC, CA, and AB,
respectively, such that lines AA₁, BB₁, and CC₁ intersect in one point P. Prove
that P is the centroid of if and only if it is the centroid of