📐 geometry

1. **Nêu bài toán:** Cho tam giác ABC với AB < AC. Tia phân giác của góc BAC cắt BC tại D. Trên AC lấy M sao cho AM = AB. 2. **Chứng minh △ABD = △AMD:**

1. Problem: Find the center and radius of a circle given by the equation $$x^2 + y^2 - 6x + 8y + 9 = 0$$. 2. Formula: The general form of a circle is $$x^2 + y^2 + Dx + Ey + F = 0$

1. **Problem statement:** Given a circle with center O, points A, B, C, and D on the circumference, and \(\angle BOC = 120^\circ\), find the value of \(x\), where \(x\) is an angle

1. **Problem Statement:** Given a parallelogram PQRS with points T and M inside it such that PT = MR and PT \parallel MR, prove:

1. **הבעיה:** יש לנו משולש שווה צלעות $ABC$ ומשולש ישר זווית ושווה שוקיים $ADC$ עם $\angle DAC=90^\circ$. נתון $DC=AB-1$ ונקודת $K$ נמצאת על $DC$. יש להוכיח את הטענות הבאות ולבטא א

1. **Problem statement:** We have an isosceles triangle ABC with vertex B as the apex, and angle ÂBC = 96°. Point I lies inside triangle ABC such that angle ÎAC = 18° and angle ÎCA

1. **Nêu bài toán:** Cho tam giác GEM vuông tại M, với $EM = \frac{1}{2} EG$. Tìm các góc $\angle MEG$ và $\angle EGM$. 2. **Công thức và quy tắc:** Trong tam giác vuông, theo định

1. **Problem statement:** Given an isosceles triangle $BCD$ with vertex at $B$, points $E$ on $BC$ and $F$ on the ray opposite to $DB$ such that $CE = DF$. Lines $EH$ and $FG$ are

1. The problem asks to compile a list of angle properties and characteristics, focusing on symmetry and angle properties. 2. Angle properties include:

1. **State the problem:** Find the surface area of a rectangular pyramid with length $l=13$ cm, width $w=11$ cm, and height $h=8$ cm. 2. **Formula and explanation:** The surface ar

1. **State the problem:** Find the surface area of a rectangular pyramid with length $l=8$, width $w=4$, and height $h=2$. The surface area includes the base area plus the area of

1. **Problem Statement:** Find the length of $\overline{BL}$ given that $\overline{TS}$ is parallel to $\overline{AL}$, with $AL=12$, $TS=7.75$, and $SL=5.1$.

1. **Problem Statement:** Given a parallelogram PQRS with points T and M inside it such that $PT = MR$ and $PT \parallel MR$, prove a certain property (usually this type of problem

1. **State the problem:** We are given a sector OAB of a circle with radius $r=5.9$ mm and area $28$ mm$^2$. We need to find the central angle $\theta$ in degrees, rounded to 1 dec

1. The problem is to determine if line segments CD and EF are coplanar. 2. Two line segments are coplanar if they lie on the same plane.

1. **Problem Statement:** We have two index cards labeled with points A, B, C, D, E, F, G, M, N and lines AB, CD, EF, and planes M and N. We need to find intersections of lines and

1. **State the problem:** We have an isosceles triangle with two equal sides each measuring 11.52 inches and a vertex angle of 32° 15'. We need to find the length of the base.

1. **State the problem:** We have a circle with center G and points I, V, E, N on the circumference. Given arcs: - Arc VE = $3x - 5$

1. **Stating the problem:** We need to find the values of angles $x$, $y$, and $z$ given some angles and relationships. 2. **Analyzing the given information:** The problem shows an

1. **State the problem:** We have two similar triangles $ABC$ and $PQR$ with sides $AB=5.5$, $BC=5$, $CA=2.5$, and $PQ=16.5$, $QR=15$, $PR=n$. We need to find $n$. 2. **Recall the

1. **State the problem:** We have a solid hemisphere H with radius $x$ cm and volume $6174\pi$ cm³. A bowl is made by removing a smaller solid hemisphere from H, leaving a uniform