📐 geometry

1. **Problem statement:** Given a right isosceles triangle $\triangle ABC$ with the right angle at $A$, $M$ is the midpoint of $BC$, and $E$ lies on segment $MC$. Lines $BH$ and $C

1. **Problem Statement:** Determine if triangles ABC and DEF are similar given some side lengths and angles, and if possible, find the length of side EF. 2. **Similarity Criteria:*

1. **Problem:** Find the nets of the following solids: cube, cuboid, prism, cylinder, cone, and sphere. 2. **Understanding Nets:** A net is a two-dimensional shape that can be fold

1. The problem states that quadrilateral PQMN lies on one circle and quadrilateral SRMN lies on another circle. 2. This means PQMN and SRMN are cyclic quadrilaterals.

1. **Problem Statement:** We are given two circles intersecting at points $M$ and $N$. Points $P$, $Q$, $R$, and $S$ lie such that $PQRS$ is a straight line. Two triangles $PNR$ an

1. مسئله: نشان دهید مثلث‌های CDE و ABC متشابه هستند و سپس طول ضلع BC را با توجه به داده‌ها محاسبه کنید. 2. برای اثبات تشابه مثلث‌ها، از قانون تشابه مثلث‌ها استفاده می‌کنیم که می‌گو

1. مسئله: در مثلث قائم الزاویه با داده‌های $AC=6$ و $HC=4$، باید طول‌های $BC$ و $AB$ را پیدا کنیم. 2. ابتدا باید بدانیم نقطه $H$ روی کدام ضلع قرار دارد. فرض می‌کنیم $H$ روی ضلع $AC

1. **Problem Statement:** Given a set of vertices and distances between them, we want to analyze the relationships or possibly find missing distances or verify consistency. 2. **Kn

1. **Stating the problem:** We have a rectangle ABCD with points E and I outside it. Given lengths are $BE=4$ cm and $AI=8$ cm. We want to analyze the figure and find relationships

1. **State the problem:** Calculate the volume of a cone with base radius $r=5$ cm and curved surface area $A=109.9$ cm$^2$. 2. **Recall formulas:**

1. **Problem Statement:** We have a circle with center $P$ and points $A, B, C, D, E$ on the circle in clockwise order. $\overline{AD}$ and $\overline{BE}$ are diameters. We want t

1. **Problem statement:** Given a semicircle $(O; R)$ with diameter $AB$, $C$ is the midpoint of $AO$. The line through $C$ perpendicular to $AO$ intersects the semicircle at $I$.

1. **Problem statement:** We have two isosceles triangles TSV and UTV with given angles and side equalities. We need to find the values of angles $x$ and $y$. 2. **Given informatio

1. **Énoncé du problème :** Soient ABC un triangle dans le plan (P) et G = bary{(A, 4); (B, 3); (C, 2)}.

1. **Problem statement:** Given circle $(O)$ and point $A$ outside it, with tangents $AB$ and $AC$ touching $(O)$ at $B$ and $C$. $H$ is the intersection of $OA$ and $BC$. 2. **Par

1. **Stating the problem:** We are given four polygons with some angles labeled and asked which angle among \(\angle a, \angle b, \angle c, \angle d\) cannot be found given the ang

1. **Problem statement:** Given a circle $(O)$ and a point $A$ outside the circle. From $A$, two tangents $AB$ and $AC$ are drawn to the circle, touching at points $B$ and $C$ resp

1. **Menentukan volume dan luas permukaan bangun 6 (prisma segiempat dengan prisma segitiga di atasnya)** Diketahui: panjang = 8 cm, lebar = 7 cm, tinggi prisma segiempat = 6 cm, t

1. **Problem statement:** Given right triangle $\triangle ABC$ right-angled at $A$ with $AB < AC$. $M$ is the midpoint of $BC$. $E$ and $F$ are the perpendicular projections of $M$

1. Bạn hỏi liệu tôi có thể giải toán hình và vẽ toán hình không. 2. Tôi có thể giúp bạn giải các bài toán hình học bằng cách sử dụng các công thức, định lý và giải thích chi tiết t

1. **Énoncé du problème :** Nous avons un triangle ABC avec I milieu de [BC]. Le point G est défini comme barycentre $G = \mathrm{Bar} \{(A,-2);(B,1);(C,-1)\}$.