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Questions related from Liudmyla Hetmanenko
I apologize to you all! The question was asked incorrectly—my mistake. Now everything is correct: In a circle with center O, chords AB and CD are drawn, intersecting at point P. In each segment...
03 October 2024 505 60 View
In triangle ∆ABC (with ∠C = 90°), the angle CBA is equal to 2α. A line AD is drawn to the leg BC at an angle α (∠BAD = α). The length of the hypotenuse is 6, and the segment CD is equal to 3....
24 September 2024 2,874 9 View
Given: In an isosceles triangle, the lengths of the two equal sides are each 1, and the base of the triangle is m. A circle is circumscribed around the triangle. Find the chord of the circle...
04 September 2024 4,772 6 View
In triangle ABC, the median BM_2 intersects the bisector AL_1 at point P. The side BC is divided by the base of the bisector AL_1 into segments CL_1=m and BL_1=n. Determine the ratio of the...
21 August 2024 7,819 4 View
In the isosceles triangle ABC (AC=AB), the angle at the vertex is 20°. Point D is chosen on the side AB such that AD=BC. Find the measure of angle CDB.
24 June 2024 8,145 8 View
In triangle ABC, the bisector AL₁ is drawn. Points O₁, O₂, O are the centers of the circles circumscribed around triangles ACL₁, ABL₁, ABC, respectively. The radii are denoted as R₁, R₂, R for...
13 June 2024 7,751 1 View
Consider a circle of radius R with center O. Two other circles are internally tangent to this circle and intersect at points A and B. Find the sum of the radii of the other two circles, given...
30 May 2024 9,937 1 View
Two circles with radii R and r touch externally at point A. Through point B, which belongs to the larger circle, a tangent to the smaller circle is drawn to point C. Find the length of the...
23 May 2024 4,924 1 View
The circle touches AB and AC the lateral sides of the isosceles triangle ABC at the vertices B and C (Fig. 1). On the arc of this circle, which lies inside this triangle, there is a point K so...
12 May 2024 7,239 6 View
In a right-angled triangle ABC (∠C=90°), the height CD equals to h is drawn. The points M and N are chosen on the continuation of AB such that NA=AD and DB=BM (Fig. 1). Find the distance from...
30 April 2024 9,025 3 View
In a right-angled triangle ABC (∠C=90°), the height CD equal to h is drawn. The points M and N are the midpoints of the segments BD and AD. Find the distance from the point C to the orthocenter...
07 April 2024 5,343 6 View
Point X belongs to the circle inscribed in the equilateral triangle ABC (see Figure 1). Lines parallel to the sides of the triangle are drawn through point X. The areas of the formed triangles...
31 March 2024 5,526 5 View
A circle is inscribed in triangle ABC (AC=AB). The point of heights’ intersection (orthocenter) (Fig. 1) belongs to this circle. Find the angles of triangle ABC (Fig. 2).
20 March 2024 1,276 8 View
In triangle ABC (with angle C = 90 degrees), the legs are equal to 5 and 7. From vertex C, segment CD is drawn (with D belonging to side BA) such that circles inscribed in triangles CAD and CBD...
27 February 2024 5,481 30 View
Construct triangle ABC using points I_b,I_c,L_1
16 February 2024 5,707 2 View
The area of the rectangle ABCD is 10. Points are chosen on the sides of the rectangle such that BM = MC, AN = NP = PD. Diagonal AC intersects NM at point K and MP at point L. Find the area of...
03 February 2024 6,612 8 View
In the square ABCD, point M is chosen inside such that ∠MAC = ∠MCD = α (see pic). Determine the measure of ∠ABM.
27 January 2024 9,910 3 View
I - the intersection point of the internal bisectors of triangle ABC; I_b - the center of the circumcircle tangent to side AC and the extensions of sides AB and BC of triangle ABC; I_c - the...
14 January 2024 6,934 3 View
In triangle ABC, the inscribed circle touches sides BC, AC, and AB at points K₁, K₂, and K₃, respectively. The excircle (Iₐ, rₐ=IₐT₂) touches the lines BC, AC, and AB at points T₁, T₂, and T₃,...
26 December 2023 5,035 9 View
In triangle ABC, the angle bisector AD of angle BAC is drawn. AC+CD=m. AB-BD=n. Find AD
19 December 2023 8,527 1 View
Dear colleagues I have a new interesting problem, and it's not easy at all! Would you like to try solving it? In the square ABCD inscribed in a circle with center O and radius 1, let X be an...
15 December 2023 8,787 24 View
Dear colleagues, I have a new original geometry problem. Please try to solve it and evaluate the author's idea: In a semicircle, a quadrilateral ABCD is inscribed such that AD is the diameter of...
04 December 2023 7,972 21 View
Dear colleagues and fans of geometry, I suggest you spend some free time and try to solve a new problem: The diagonals of the quadrilateral ABCD inscribed in a circle with center O are...
27 November 2023 7,518 4 View
Dear colleagues and geometry enthusiasts, I invite you to tackle an original aesthetic geometry problem: A semicircle is circumscribed on the side of square ABCD, acting as its diameter. Find:...
18 November 2023 9,908 3 View
Here is a new problem for fans of geometric constructions. Try to solve it! Given: AB+BD=AC+CD (see pic). Determine the type of trapezoid ABCD.
15 November 2023 6,620 7 View
Dear colleagues and fans of geometry, I suggest you spend some free time and try to solve a new problem: Is it possible to construct a triangle ABC with only this data: r_b; r_c; |∠B-∠C|/2?...
04 November 2023 4,785 3 View
Do lines t₁, t₂, and t₃ intersect at a single point? If yes, prove it! Please, take a look at the problem statement in the attached photos.
30 October 2023 213 2 View
Dear colleagues and fans of geometry, I suggest you spend some free time and try to solve a new problem: reconstruct triangle ABC from point A, S, F, M1 (CM1=M1B)
11 October 2023 8,715 3 View
Dear colleagues, I have a new original geometry problem. Please try to solve it and evaluate the author's idea: Reconstruct triangle ABC from points A, E, F, and H1 (see the picture).)
04 October 2023 5,984 4 View
Dear colleagues and enthusiasts of beautiful geometric problems, I invite you to solve another elegant problem: Reconstruct triangle ABC from points A, D, E, and H1. I will be glad to see your...
27 September 2023 9,869 4 View
In the 'Collection of Geometric Problems' from 1966, there is a problem in which the author made a mistake. Try to find the author's error! In the picture, you can see the conditions of this...
20 September 2023 7,118 1 View
Dear colleagues and fans of #emotionalgeometry, I suggest you spend some free time and try to solve a beautiful and, at first glance, not difficult problem: reconstruct triangle ABC from point O...
13 September 2023 211 3 View
Dear colleagues and geometry enthusiasts, I invite you to tackle an original aesthetic geometry problem: Reconstruct triangle ABC using points M1, L1, and H2 (based on the diagram).
04 September 2023 1,712 15 View
