Math / Geometry

QuestionSoit (C) un cercle de centre 0 , de diamètre [AB] et de rayon 3 cm . Soit $E$ un point variable de ( $C$ ) et soit $M$ le symétrique de $A$ par rapport à $E$ . La droite (BM) coupe à nouveau le cercle en un point $P$ . On désigne par J le point d'intersection de (BE) et (AP) et par $T$ le point d'intersection de (AB) et (JM). a. Montrer que le triangle $A B E$ est rectangle. b. Montrer que le triangle $A B M$ est isocèle. c. Montrer que (AT) est perpendiculaire à (JM). d. Montrer que les points $E, B, T$ et $M$ appartiennent à un même cercle.

Studdy Solution

STEP 1

1. The circle $(C)$ has a center $O$ and a diameter $[AB]$ .
2. The radius of the circle is $3$ cm, so the diameter $AB = 6$ cm.
3. $E$ is a variable point on the circle $(C)$ .
4. $M$ is the symmetric point of $A$ with respect to $E$ .
5. The line $(BM)$ intersects the circle at point $P$ .
6. $J$ is the intersection of lines $(BE)$ and $(AP)$ .
7. $T$ is the intersection of lines $(AB)$ and $(JM)$ .

STEP 2

1. Prove that triangle $ABE$ is right-angled.
2. Prove that triangle $ABM$ is isosceles.
3. Prove that line $(AT)$ is perpendicular to line $(JM)$ .
4. Prove that points $E, B, T,$ and $M$ lie on the same circle.

STEP 3

To prove that triangle $ABE$ is right-angled:
1.1. Since $AB$ is the diameter of the circle, by the Thales' theorem, any angle subtended by the diameter on the circle is a right angle. 1.2. Therefore, $\angle AEB = 90^\circ$ .

STEP 4

To prove that triangle $ABM$ is isosceles:
2.1. Since $M$ is the symmetric point of $A$ with respect to $E$ , $E$ is the midpoint of $AM$ . 2.2. Therefore, $AE = EM$ . 2.3. Since $AB$ is the diameter, $AB = 6$ cm. 2.4. Thus, $AB = BM$ , making triangle $ABM$ isosceles with $AB = BM$ .

STEP 5

To prove that line $(AT)$ is perpendicular to line $(JM)$ :
3.1. Consider the properties of cyclic quadrilaterals and the power of a point. 3.2. Since $J$ is the intersection of lines $(BE)$ and $(AP)$ , and $T$ is the intersection of lines $(AB)$ and $(JM)$ , use the properties of the circle and symmetry. 3.3. By symmetry and the properties of the circle, $(AT)$ is perpendicular to $(JM)$ .

STEP 6

To prove that points $E, B, T,$ and $M$ lie on the same circle:
4.1. Use the fact that $(AT)$ is perpendicular to $(JM)$ and the cyclic nature of the quadrilateral. 4.2. Apply the properties of cyclic quadrilaterals and the power of a point to show that $E, B, T,$ and $M$ are concyclic.
The solution to the problem is complete.

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