Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 4.8 AIME Find the answer

5. Among the following propositions:
(1) A quadrilateral that is both axially symmetric and centrally symmetric is a rhombus;
(2) In $\triangle A B C$ and $\triangle D E F$ , $\angle A=\angle D=$ $99^{\circ}, A B=D E=12, B C=E F=20$ . Then $\triangle A B C$ and $\triangle D E F$ are congruent;
(3) If three lines intersect two lines, and the corresponding segments formed are proportional, then these three lines are parallel;
(4) If the orthocenter of a triangle lies on one of its medians, then the triangle is isosceles.
Which of the following are true propositions?:

Pick one

Solution

5. (D).
(1) The quadrilateral that satisfies the conditions is a rectangle or a rhombus.
(2) Use a ruler and compass for construction.
(3) It is easy to find a counterexample through construction.
(4) The orthocenter of a right triangle is the right-angle vertex and also lies on the median.

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.