CLASSICAL THEOREMS IN PLANE GEOMETRY. September 2007 BC 1 AC 1 CA 1 BA 1 = 1.

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1 SSI THEORES I E GEOETRY ZVEZEI STKOV September 2007 ote: ll 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, and are collinear, then the directed ratio / is the ratio of the lengths of segments and, taken with a sign + if the segments have the same direction (i.e. is not between and ), and with a sign if the segments have opposite directions (i.e. is between and ). Several objects (lines, circles, etc.) are concurrent if they all intersect in some point.. (enelaus) et, and be three points on the sides, and of. rove that they are collinear (cf. Fig. ) iff =. Figure Figure 2a 2. (a) rove 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 collinear points. (cf. Fig. 2a) (b) rove that the exterior angle bisectors of nonisosceles intersect the continuations of opposite sides of in three collinear points. (c) rove 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. (ascal) If the hexagon EF is cyclic and its opposite sides, and E, and EF, and F, 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 These all are conics, i.e. projectively equivalent to a circle.

2 2 ZVEZEI STKOV F E X Y Figure 2c Figure 3 Z then replace the corresponding side of the hexagon by the tangent to the circle at the corresponding vertex. Thus, obtain the following: (a) If =, =, = F, deduce to roblem 2c. (b) If E = F, formulate the property of any inscribed pentagon. (c) If = F and = E, for the inscribed quadrilateral we have: the intersection points of and the tangent at, of and the tangent at, and of and, are collinear. (d) If = F and =, 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. (ctually, the tangents at any pair of opposite vertices should also work.) 4. (esargues) and are positioned in such a way that lines,, and intersect in a point O. If lines and, and, and are pairwise not parallel, prove that their points of intersection,, and, are collinear. (cf. Fig. 4) H O K I T Figure 4 Figure 5 ote: 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 is the circle tangent to side and to the extentions of sides and. Its center is called an excenter of. On what bisectors does this excenter lie? The circumcircle of is the circle passing through the vertices, and. Its center is called the circumcenter of ; it lies on the perpendicular bisectors of the sides of the triangle.

3 SSI THEORES I E GEOETRY 3 5. rove that the midpoint K of the altitude H in, the incenter I of, and the tangency point T on of the excircle of (tangent to side ) are collinear. (cf. Fig. 5) 6. (Gauss s line with respect to l) ine l intersects the sides (or continuations of), and of in points, 2 and 3. rove that the midpoints, 2 and 3 of, 2 and 3 are collinear. (cf. Fig. 6) l Figure 6 Figure et be a quadrilateral with perpendicular diagonals intersecting in. The feet of the perpendiculars from to sides,, and are, 2, 3 and 4. rove that lines 2, 3 4 and are concurrent. (cf. Fig. 7) 8. (Simpson) rove that the feet of the perpendiculars dropped from a point on the circumcircle k of to the sides of the triangle are collinear. ore generally, let S be the area of, R the circumradius, and d the radius of a circle ɛ concentric to k. et, and be the feet of the perpendiculars dropped from an arbitrary point on ɛ to the sides of. rove that the area S of is given by the formula S = 4 S d2. In particular, when ɛ = k, then S R 2 = 0, and hence, and are collinear. (cf. Fig. 8) Figure 8 9. (Salmon) Through a point on a circle ɛ draw three arbitrary chords, and, and using each chord as a diameter, draw three new circles ɛ, ɛ 2, and ɛ 3. rove that the pairwise intersections of the ɛ i s (other than ) are collinear. ote: et 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 H, H and H, and the feet of the altitudes of. In fact, the

4 4 ZVEZEI STKOV 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. 0a) H E O H E Figure 0a Figure 0b 0. rove that Simpson s line of with respect to point on the circumscribed circle k of, line H where H is the orthocenter of, and the Euler circle of 9 points for are concurrent. (cf. Fig. 0b). (eva) et, and be three points on the sides (or continuations of),, of. rove that,, are concurrent or are parallel iff =. 2. (Gergonne s point) rove that the lines connecting the vertices of a triangle with the points of tangency of the inscribed circle are concurrent. (cf. Fig. 2) G Figure 2 Figure 3 3. (agel s point) rove 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. 3.) ote that these are also the three lines through the vertices of the triangle and dividing each its perimeter into two equal parts. 4. et be an arbitrary point on side of. et and Q be the intersection points of the angle bisectors of and with sides and, respectively. rove that lines, Q and are concurrent.

5 SSI THEORES I E GEOETRY 5 5. et,, be points on the sides of an acuteangled so that the lines, and are concurrent. rove that is an altitude in iff it is the angle bisector of. 6. In the acuteangled a semicircle k with center O on side is inscribed. et and be the points of tangency of k with sides and. rove that lines, and the altitude of are concurrent. (cf. Fig. 6) 7. circle k intersects side of in and 2, side in and 2, side in and 2. The order of these points on k is:, 2,, 2, 2,. rove that lines,, are concurrent iff 2, 2, 2 are concurrent. (cf. Fig. 7) 2 2 I O 2 Figure 6 Figure 7 Figure 8 8. et the points of tangency of the incircle of with the sides, and be, and, and let 2, 2 and 2 be their reflections across the incenter I of. rove that lines 2, 2 and 2 are concurrent. (cf. Fig. 8) 9. (Gauss) If the two pairs of opposite sides of a quadrilateral intersect in E and F, prove that the midpoint of EF lies on the line through the midpoints and of the diagonals and. (cf. Fig. 9) F E Figure 9 Figure oint lies inside. ines,, intersect the sides,, in,,, respectively, and,,,,, are the midpoints of the segments,,,,,. rove that, and are concurrent. (cf. Fig. 20)

6 6 ZVEZEI STKOV O 3 Q I O O O2 R Figure 2 Figure et, Q and R be points on the sides, and of. et O, O 2 and O 3 be the circumcenters of QR, R and Q. rove that O O 2 O 3. (cf. Fig. 2) 22. (IO 8) Three congruent circles pass through point inside. Each circle is inside and is tangent to two of its sides. rove that the circumcenter O and incenter I of and are collinear. (cf. Fig. 22) 23. (rianchon) If the hexagon EF is circumscribed around a circle, prove that its three diagonals, E and F are concurrent. (cf. Fig. 23) E F 3 I 3 Figure 23 Figure (Saint etersburg Olympiad) oint I is the incenter of. Some circle with center I intersects side in and 2, side in and 2, and side in and 2. The six points obtained in this way lie on the circle in the following order:, 2,, 2,, 2. oints 3, 3 and 3 are the midpoints of the arc 2, 2 and 2 respectively. ines 2 3 and 3 intersect in 4, lines 2 3 and 3 in 4, and lines 2 3 and 3 in 4. rove that the segments 3 4, 3 4 and 3 4 intersect in one point. (cf. Fig. 24) 25. (ulgarian IO Test 08) In let ( ) be a median and let ( ) and ( ) be two altitudes. The line through perpendicular to intersects lines and in points E and F, respectively. enote by k the circumcircle of EF. et k and k 2 be circles tangent to both EF and to the arc EF on K not containing. If and Q are the intersection points of k and k 2, prove that points, Q, and lie on a line.

7 SSI THEORES I E GEOETRY (IO 08, Spain) n acute-angled has orthocenter H. The circle passing through H with center the midpoint of intersects the line at and 2. Similarly, the circle passing through H with center the midpoint of intersects the line at and 2, and the circle passing through H with center the midpoint of intersects the line at and 2. Show that, 2,, 2,, and 2 lie on a circle. 27. (O 06) Given, let,, and be points on sides,, and, respectively, such that lines,, and intersect in one point. rove that is the centroid of if and only if it is the centroid of. Zvezdelina Stankova, ills ollege, Oakland,, erkeley ath ircle irector, stankova@math.berkeley.edu