Basic Geometry: Refresher for pre-RMO, RMO and IITJEE foundation maths

1) Assume that in a $\bigtriangleup$ ABC and $\bigtriangleup RST$, we know that $AB=RS$, $AC=RT$, $BC=ST$. Prove that $\bigtriangleup ABC \cong \bigtriangleup RST$ without using the SSS congruence criterion.

2) Let $\bigtriangleup ABC$ be isosceles with base BC. Then, $\angle B = \angle C$. Also, the median from vertex A, the bisector of $\angle A$, and the altitude from vertex A are all the same line. Prove this.

3) If two triangles have equal hypotenuses and an arm of one of the triangles equals an arm of the other, then the triangles are congruent. Prove.

4) An exterior angle of a triangle equals the sum of the two remote interior angles. Also, the sum of all three interior angles of a triangle is 180 degrees.

5) Find a formula for the interior angles of an n-gon.

6) Prove that the opposite sides of a parallelogram are equal.

7) In a quadrilateral ABCD, suppose that AB=CD and AD=BC. Then, prove that ABCD is a parallelogram.

8) In a quadrilateral ABCD, suppose that AB=CD and AB is parallel to CD. Then, prove that ABCD is a parallelogram.

9) Prove that a quadrilateral is a parallelogram iff its diagonals bisect each other.

10) Given a line segment BC, the locus of all points equidistant from B and C is the perpendicular bisector of the segment. Prove.

11) Corollary to problem 10 above: The diagonals of a rhombus are perpendicular. Prove.

12) Let AX be the bisector of $\angle A$ in $\triangle ABC$. Then, prove $\frac{BX}{XC} = \frac{AB}{AC}$. In other words, X divides BC into pieces proportional to the lengths of the nearer sides of the triangle. Prove.

13) Suppose that in $\triangle ABC$, the median from vertex A and the bisector of $\angle A$ are the same line. Show that $AB=AC$.

14) Prove that there is exactly one circle through any three given non collinear points.

15) An inscribed angle in a circle is equal in degrees to one half its subtended arc. Equivalently, the arc subtended by an inscribed angle is measured by twice the angle. Prove.

16) Corollary to above problem 15: Opposite angles of an inscribed quadrilateral are supplementary. Prove this.

17) Another corollary to above problem 15: The angle between two secants drawn to a circle from an exterior point is equal in degrees to half the difference of the two subtended arcs. Prove this.

18) A third corollary to above problem 15: The angle between two chords that intersect in the interior of a circle is equal in degrees to half the sum of the two subtended arcs. Prove this.

19) Theorem (Pythagoras): If a right triangle has arms of lengths a and b and its hypotenuse has length c, then $a^{2}+b^{2}=c^{2}$. Prove this.

20) Corollary to above theorem: Given a triangle ABC, the angle at vertex C is a right angle iff side AB is a diameter of the circumcircle. Prove this.

21) Theorem: The angle between a chord and the tangent at one of its endpoints is equal in degrees to half the subtended arc. Prove.

22) Corollary to problem 21: The angle between a secant and a tangent meeting at a point outside a circle is equal in degrees to half the difference of the subtended arcs.

23) Fix an integer, $n \geq 3$. Given a circle, how should n points on this circle be chosen so as to maximize the area of the corresponding n-gon?

24) Theorem: Given $\bigtriangleup ABC$ and $\bigtriangleup XYZ$, suppose that $\angle A = \angle X$ and $\angle B= \angle Y$. Then, prove that $\angle C = \angle Z$ and so $\bigtriangleup ABC \sim \bigtriangleup XYZ$. Prove this theorem.

25) Theorem: If $\bigtriangleup ABC \sim \bigtriangleup XYZ$, then the lengths of the corresponding sides of these two triangles are proportional. Prove.

26) The following lemma is important to prove the above theorem: Let U and V be points on sides AB and AC of $\bigtriangleup ABC$. Then, UV is parallel to BC if and only if $\frac{AU}{AB} = \frac{AV}{AC}$. You will have to prove this lemma as a part of the above proof.

27) Special case of above lemma: Let U and V be the midpoints of sides AB and AC, respectively in $\bigtriangleup ABC$. Then, UV is parallel to BC and $UV = \frac{1}{2}BC$.

28) Suppose that the sides of $\bigtriangleup ABC$ are proportional to the corresponding sides of $\bigtriangleup XYZ$. Then, $\bigtriangleup ABC \sim \bigtriangleup XYZ$.

29) Given $\bigtriangleup ABC$ and $\bigtriangleup XYZ$, assume that $\angle X = \angle A$ and that $\frac{XY}{AB} = \frac{XZ}{AC}$. Then, $\bigtriangleup ABC \sim \bigtriangleup XYZ$.

30) Consider a non-trivial plane geometry question now: Let P be a point outside of parallelogram ABCD and $\angle PAB = \angle PCB$. Prove that $\angle APD = \angle CPB$.

31) Given a circle and a point P not on the circle, choose an arbitrary line through P, meeting the circle at points X and Y. Then, the quantity $PX.PY$ depends only on the point P and is independent of the choice of the line through P.

32) You can given an alternative proof of Pythagoras’s theorem based on the following lemma: Suppose $\bigtriangleup ABC$ is a right triangle with hypotenuse AB and let CP be the altitude drawn to the hypotenuse. Then, $\bigtriangleup ACP \sim \bigtriangleup ABC \sim \bigtriangleup CBP$. Prove both the lemma and based on it produce an alternative proof of Pythagorean theorem.

33) Prove the following: The three perpendicular bisectors of the sides of a triangle are concurrent at the circumcenter of the triangle.

34) Prove the law of sines.

35) Let R and K denote the circumradius and area of $\bigtriangleup ABC$, respectively and let a, b and c denote the side lengths, as usual. Then, $4KR = abc$.

36) Theorem: The three medians of an arbitrary triangle are concurrent at a point that lies two thirds of the way along each median from the vertex of the triangle toward the midpoint of the opposite side.

37) Time to ponder: Prove: Suppose that in $\bigtriangleup ABC$, medians BY and CZ have equal lengths. Prove that $AB=AC$.

38) If the circumcenter and the centroid of a triangle coincide, then the triangle must be equilateral. Prove this fact.

39) Assume that $\bigtriangleup ABC$ is not equilateral and let G and O be its centroid and circumcentre respectively. Let H be the point on the Euler line GO that lies on the opposite side of G from O and such that $HG = 2GO$. Then, prove that all the three altitudes of $\bigtriangleup ABC$ pass through H.

40) Prove the following basic fact about pedal triangles: The pedal triangles of each of the four triangles determined by an orthic quadruple are all the same.

41) Prove the following theorem: Given any triangle, all of the following points lie on a common circle: the three feet of the altitudes, the three midpoints of the sides, and the three Euler points. Furthermore, each of the line segments joining an Euler point to the midpoint of the opposite side is a diameter of this circle.

42) Prove the following theorem and its corollary: Let R be the circumradius of triangle ABC. Then, the distance from each Euler point of $\bigtriangleup ABC$ to the midpoint of the opposite side is R, and the radius of the nine-point circle of $\bigtriangleup ABC$ is $R/2$. The corollary says: Suppose $\bigtriangleup ABC$ is not a right angled triangle and let H be its orthocentre. Then, $\bigtriangleup ABC$, $\bigtriangleup HBC$, $\bigtriangleup AHC$, and $\bigtriangleup ABH$ have equal circumradii.

43) Prove the law of cosines.

44) Prove Heron’s formula.

45) Express the circumradius R of $\bigtriangleup ABC$ in terms of the lengths of the sides.

46) Prove that the three angle bisectors of a triangle are concurrent at a point I, equidistant from the sides of the triangle. If we denote the by r the distance from I to each of the sides, then the circle of radius r centered at I is the unique circle inscribed in the given triangle. Note that in order to prove this, the following elementary lemma is required to be proved: The bisector of angle ABC is the locus of points P in the interior of the angle that are equidistant from the sides of the triangle.

47) Given a triangle with area K, semiperimeter s, and inradius r, prove that $rs=K$. Use this to express r in terms of the lengths of the sides of the triangle.

Please be aware that the above set of questions is almost like almost like a necessary set of pre-requisites for RMO geometry. You have to master the basics first.

Regards,

Nalin Pithwa.

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