Very hard geometry problem for 9 th grade student

Nội dung text: Very hard geometry problem for 9 th grade student

1. Very hard geometry problem for 9 th grade student Introduction: In this text I will give a very difficult level of geometry problems about circles. Please consult and try these plane geometry problems together. Hope will bring many useful things and skills to solve difficult geometry problems for students Problem 1: Let an acute triangle ABC (AB < AC) have 2 heights BE and CF. The point M of the line BE, the point N is on the line CF such that AB _|_ AM and AC _|_ AN. Let EF intersect MN at I. Prove: AI is the tangent of the circumcircle of the triangle AEF Problem 2: Let an acute triangle ABC (AB < AC) have two heights BE and CF intersect at H. Let EF intersect (O) at M and N (MF < ME). Let I be the midpoint of the side BC, and MI intersects (O) at K. Prove: AK is perpendicular to HN Problem 3: Let an acute triangle ABC (AB <AC) have three heights AD, BE and CF intersect at H. Let EF intersect (O) at M and N (MF < ME). Prove that: a/ EF = b/ Problem 4: From 1 point A outside the circle (O; R). Draw two lines tangent to (O) at B and C. A straight line goes through A intersects (O;R) at M and N such that AM < AN and M with C lie on the other side of the line OA. Let OA intersect with BC at H and BC intersect with MN at I. Prove: a/ b/ If OA = 2R and I is the midpoint of the side BH. Calculate the area of the quadrilateral BNCM according to R Problem 5: Let the equilateral triangle ABC be inscribed with the circle (O; R). The point M belongs to the line segment AC such that MA = 2MC. Draw MN perpendicular to AB at N, MN intersects (O) at I (IN < IM). a/ Calculate the length IN according to R
2. b/ Calculate the area of triangles AOI and IBC according to R Problem 6: The acute-pointed triangle ABC is inscribed in a circle (O; R) with AB <AC. The tangents at B and C intersect at D, AD intersect with BC at E and intersect circle (O) at I. The circumcircle of the triangle AEC intersects with AB at the second point is N. Let M be the symmetry point N through E. Prove: The quadrilateral ANIM is inscribed in a circle Problem 7: Point A is outside the circle (O). From A draw two lines tangent to (O) at B and C. Any point M belongs to the small arc of the circle BC of (O). The line through A parallel to BC intersects CM and BM at E and F. Prove: When M is move on the small arc circle BC, the orthocentre of the triangle OEF is a fixed point Problem 8: Given an acute triangle ABC (AB <AC) inscribed in a circle (O; R) with 2 heights BE and CF intersecting at H, AH intersects (O) at point D. The line through H parallel to EF intersects the line BC at I. Let AI intersect (O) at S. Draw the diameter AK. Prove: 3 lines BC, OD, SK are concurrent Problem 9: Let an acute triangle ABC (AB <AC) inscribed in a circle (O; R) have 2 heights AD and BE. Let K be the midpoint of side BC and I is the symmetry point E across the line BC. The line going through I and perpendicular to IK intersects the line AD at point S. Prove: OK = DA – DS Problem 10: From a point A outside the circle (O; R), draw 2 lines tangent to (O) at B and C. Draw the diameter CD, AD intersect (O) at E. Let I be the midpoint next to DE, BD intersects with EC at point F, IF intersects with OA at point S. Prove: SD is tangent to circle (O) Problem 11: From a point A outside the circle (O; R), draw 2 lines tangent to (O) at B and C. Draw the diameter CD, AD intersect (O) at E. The line passing through A and parallel to BE intersects the line BC at K. Let OA intersect BC at H, the line HN is perpendicular to CD at N and KM is perpendicular to AD at M. Prove: The quadrilateral BMNH is a parallelogram Problem 12: From a point A outside the circle (O; R), draw 2 lines tangent to (O) at B and C. Draw the diameter CD, OA intersect BC at H, AD intersect (O) at I, DH intersect AC at N, AC intersect BD at E. Prove: The triangle INE is a right triangle
3. Problem 13: Let an acute triangle ABC (AB < AC) be inscribed in a circle (O; R) with 2 heights BE and CF intersect at H. Let EF intersect BC at M, draw diameters AK, MH intersect KC at I. Prove: AI intersects BE at one point on the circle (O) Problem 14: From 1 point A outside the circle (O; R), draw a line tangent to (O) at B. Draw BH perpendicular to OA at H. The line OA intersects (O) at M and N (AM < AN). . Prove: Problem 15: From a point A outside the circle (O; R), draw a line tangent to (O) at B and C. A line passing through A without going through center O intersects (O) at M and N (AM < AN), M and C are on the other side of the line OA. Let E is the midpoint of side MN, EC intersects BD at I. Let AI intersect BC at K. On the line EK take point F such that OF _|_ AF. Prove: The quadrilateral BKMF is inscribed in the circle Problem 16: From a point A outside the circle (O; R), draw a line tangent to (O) at B and C. A line passing through A without going through center O intersects (O) at D and E (AD < AE), D and C are on the other side of the line OA. The circumcircle of the two triangles ODE and OBC intersect at the second point is M. Prove: 3 lines DE, BC, OM are concurrent Problem 17: From a point A outside the circle (O; R), draw a line tangent to (O) at B and C. Point D moves on the small arc BC of the circle. Let BD intersect AC at N and CD intersect AB at M. Sign: symbol S is the area. Prove: Problem 18: Given an acute triangle ABC (AB < AC) inscribed in circle (O; R). Draw the chord BF of (O) perpendicular to AC. Constructing parallelogram FBCI. Let M be the midpoint of side BC and N the midpoint of side AF. Prove: AI = 2MN Problem 19: Let an acute triangle ABC (AB < AC) be inscribed in a circle (O; R) with 2 heights BE and CF intersect at H. Let AH intersect EF at N, OA intersect BC at M. Let I be the midpoint of side MN, AI intersects NC at K. Prove: FK is parallel to BC Problem 20: Point C belongs to a circle (O; R) with diameter AB. The tangent at A of the circle intersects BC at D. Let E be the midpoint of side AD, BE intersects (O) at I. Prove: When C moves on the circle, the line passes through C and perpendicular with IC always going through a fixed point
4. Problem 21: Let ABC be an inscribed acute triangle ABC (O; R) with AB <AC. Draw the tangent line Ax at A of (O). Let E and F be the projection of B and C on the line Ax respectively. Draw AD perpendicular to BC at D. The line through E perpendicular to DE intersects AD at I. Let AD intersect (O) at K, AC intersect DF at M a/ Prove: AI = DK b/ IM intersects EC at H, KM intersects (O) at N. Prove: The quadrilateral HNFC is inscribed in the circle. Problem 22: Let the isosceles triangle ABC at A is inscribed in the circle (O; R). Draw the median BM and the height BH of the triangle ABC (M, H are on edge AC). Let BM and BH intersect the circle (O) at N and K. Calculate the ratio of the area of the quadrilateral NKHM to quadrilateral area AKNO if Problem 23: Given an acute triangle ABC (AB <AC). The center circle (O; R), diameter BC intersects AB at M and intersects AC at N, BN intersects CM at H, MN intersects BC at E. Tangent at B of (O) intersects the AE at I a/ Prove: IN is the tangent of the circle (O) b/ MI intersects AH at K, IC intersects AH at P. In case of quadrilateral CKPN is inscribed in circle and . (The symbol S is the area). Prove: the value A = has an integer leaf value Problem 24: The circle whose center (I) is inscribed with the triangle ABC is tangent to side BC at D. Take the point O of the line AD such that D lies between 2 points A and O. The line passing through I with perpendicular to AD intersects OB at E and intersect OC at F. Let M be the midpoint of side DE and N be the midpoint of side DF, BM intersect CN at K. Prove: OK passes through the midpoint of side EF Problem 25: The center circle (I) inscribed in the triangle ABC is tangent to the sides BC, AB, AC at D, E, F. Draw the diameter DN of (I), DN intersects EF at M. Construct a point L such that AL // BC and AN // ML, DL intersect EF at K. Prove: DK is a bisector of angle Problem 26: Point C is a circle (O; R) of diameter AB such that AC < BC. Draw CH perpendicular to AB at H. The tangents at A and C of (O) intersect at I, BI intersect
5. OC at E. Take the points F and K on the line BC such that EF // BC and OK // AI. Prove that: a/ BF2 = FK.FC b/ Problem 27: Point C is a circle (O; R) with diameter AB such that AC > BC. Draw CH perpendicular to AB at H. Let I is the midpoint of side BC. The tangent at B of (O) intersects OI at D, AD intersects (O) at K. Draw BG perpendicular to HK at G. a/ Prove: When C moves on (O), the line IK always passes through a fixed point b/ Prove: , then calculate the area of the triangle BCG according to R in case the value of the trigonometric tanKCG reaches the maximum value Problem 28: Let the triangle ABC be square at A with the height AH. The bisector of the angle intersects AB at D, the bisector of the angle intersects AC at E. Draw AK perpendicular to CD at K. Prove: DE is the tangent of the circumcircle of the triangle EBK Problem 29: From 1 point A outside (O; R), draw two tangent lines with (O) at B and C. Draw the diameters BD, AD intersect BC at I. Let M on the line BD, N on the line AC such that IM is perpendicular to AD and IN is parallel to AB. Prove: BC passes through the midpoint of MN. Problem 30: Let an acute triangle ABC (AB < AC) have 3 heights AD, BE, CF intersect at H, EF intersect AH at O and intersect BC at I. Prove: a/ b/ Problem 31: Let an acute triangle ABC (AB < AC) have 3 heights AD, BE, CF intersect at H. Draw CK perpendicular to EF at K. Let I is the midpoint of side AF. Prove: IDK is an isosceles triangle Problem 32: Let an acute triangle ABC (AB <AC) inscribed in a circle (O; R) with 3 heights AD, BE, CF intersect at H. Let OA intersect EF at K. Point I belongs to the line FC such that DK // EI. Prove: H is the midpoint of side IF Problem 33: Let an acute triangle ABC (AB < AC) inscribed in a circle (O; R) with 3 heights AD, BE, CF intersect at H. Let EF intersect AH at I. Construct a point K such
6. that HK // BC and CK // AB. Let BK intersect FC at S. Let M be the midpoint of side BC. Prove: a/ b/ BI // MS Problem 34: Let an acute triangle ABC (AB < AC) inscribed in a circle (O; R) with 2 heights BE and CF intersect at H. Let AH intersect (O) at K and I be the midpoint of the side AH. Let IE intersect BK at M and IF intersect CK at N. Prove: OH is perpendicular to MN Problem 35: Let an acute triangle ABC (AB < AC) inscribed in a circle (O; R) with 3 heights AD, BE, CF intersect at H. Let M and N be the projections of C and B on the line EF. Prove: Problem 36: Let an acute triangle ABC (AB < AC) inscribed in a circle (O; R) with 3 heights AD, BE, CF intersect at H. Draw DK perpendicular to EF at K. Prove: DK is the bisector of the angle Problem 37: Let an acute triangle ABC (AB < AC) inscribed in a circle (O; R) with 3 heights AD, BE, CF intersect at H. Let M and N be the projection of F and E on the line AH. Sign: S is the symbol of area. Prove: a/ b/ Problem 38: Let ABC be an acute triangle (AB < AC). The circle with center O, diameter BC intersects line AB at F and intersects line AC at E. Let BE intersect CF at H. Let I be the symmetry point H through point O. Let IF intersect BE at P, IE intersect CF at Q, PQ intersects BC at M. From M draw a line tangent to (O) at K. Prove: MK = MI Problem 39: Let the triangle ABC be square at A (AB <AC) with the height AH. The circle with diameter AH intersects the line AB at D and intersects the line AC at E. Sign: S is the symbol of area and P is the circumference 1/ Calculate the value of expressions:
7. A = B = 2/ Value representation of expressions a/ Indicate B = BC2 – BD2 – EC2 to HB and HC b/ Find the ratio if 3/ Prove that: a/ (HC + HE).(BC + AB) = (AH + AC)2 b/ BE.CD = BD.HD + EH.EC + 3BD.EC c/ d/ AB.AC.BC – BD.HB.HB – EC.EC.HC = 3BD.EC.BC e/ f/ g/ h/ k/ l/ m/ n/ Problem 40: Let the triangle ABC be square at A (AB <AC) with the height AH. Let D and E be the midpoints of AB and AC respectively. Let M and N be the projection of A on the lines BD and EC respectively. Let K be the symmetry point C through H, HN intersects AC at I. Let S be the area. Prove that:
8. a/ b/ c/ d/ BM2 + NC2 = . If BM2 + NC2 = 2.(HM2 + HN2) then the triangle ABC is isosceles e/ The 3 points K, M, I are collinear f/ Find the ratio if ^^^&&& GOOD LUCK &&&^^^