📐 14 Classical Euclidean Geometry Problems for Practice

Here is a collection of 14 classical Euclidean geometry problems from Problems and Solutions in Euclidean Geometry by M. N. Aref and William Wernick.

They involve metric relations, areas, perpendiculars, squares, quadrilaterals, minimum-distance problems and elegant identities.

The problems are presented here without solutions, for practice.

1. If \(A,B,C,D\) lie on a straight line, prove that \[ \boxed{AC\cdot BD=AB\cdot CD+BC\cdot AD}. \]

2. Two equal line segments intersect at right angles. Join their four endpoints to form a quadrilateral. Prove that the area of this quadrilateral is equal to half the area of the square constructed on either segment.

3. Let \(R\) be the midpoint of the segment \(PQ\). From \(P,Q,R\), draw perpendiculars \(PL,QM,RN\) to another straight line, meeting it at \(L,M,N\), respectively. Prove that \[ \boxed{[PLMQ]=LM\cdot RN}. \]

4. In an isosceles triangle, draw a perpendicular from one of the equal-angle vertices to the opposite side. Compare the square on this perpendicular with the square on the segment intercepted between the other equal-angle vertex and the perpendicular. Show that the difference is twice the rectangle contained by the two segments into which the corresponding side is divided, according as the vertex angle is acute or obtuse.

5. The segment \(BA\) is divided at \(C\) so that \[ AC^2=AB\cdot BC. \] The line is produced to \(D\) so that \[ AD=2AC. \] Prove that \[ \boxed{BD^2=5AB^2}. \]

6. An equilateral triangle is constructed with one vertex at \(B\), a vertex of another equilateral triangle \(ABC\), and its opposite side \(PQ\) passes through \(C\). Prove that \[ \boxed{BA\cdot BP=AB^2+PC\cdot QC}. \]

7. On a given straight line, construct a rectangle whose area is equal to the difference of the squares on two given line segments.

8. Prove that any rectangle is equal to half the rectangle contained by the diagonals of the square constructed on its two adjacent sides.

9. The points \(A\) and \(B\) are fixed, and \(CD\) is a fixed straight line of indefinite length. Find the point \(P\in CD\) for which \[ \boxed{PA^2+PB^2} \] is a minimum.

10. Prove that the sum of the squares on the diagonals of any quadrilateral is twice the sum of the squares on the segments joining the midpoints of opposite sides.

11. Prove that the sum of the squares of the distances of any point from the four vertices of a parallelogram exceeds the sum of the squares on two adjacent sides by four times the square of the distance from the point to the intersection of the diagonals.

12. Let \(ABC\) and \(DEF\) be two triangles such that \[ AB=DE,\qquad AC=DF, \] and \[ \angle BAC+\angle EDF=180^\circ. \] Prove that \[ \boxed{ BC^2+EF^2=2\left(AB^2+AC^2\right) }. \]

13. Two squares \(ABCD\) and \(A'B'C'D'\) are positioned so that corresponding sides are parallel. Prove that \[ \boxed{ AA'^2+CC'^2=BB'^2+DD'^2 }. \]

14. Squares are constructed externally on the four sides of a quadrilateral. Join adjacent corners of consecutive squares to obtain an eight-sided figure. Prove that the sum of the squares on the eight sides of this figure, together with twice the sum of the squares on the diagonals of the original quadrilateral, is equal to five times the sum of the squares on the four sides of the quadrilateral.

💡 Geometry practice

Try to solve the problems synthetically before using coordinates or vectors. Many of them were designed to reward a well-chosen auxiliary line, a classical Euclidean theorem, or an area interpretation.

📖 Read the Full Book

📚 Source

M. N. Aref & William Wernick, Problems and Solutions in Euclidean Geometry, Dover Publications, 1968.

📚

Έρχεται το πολλαπλό βιβλίο ΝΕΟ — βρες όλες τις επιλογές εδώΠολλαπλό βιβλίο ΝΕΟ — 437 βιβλία σε PDF

PDF & Ψηφιακά Μαθησιακά Αντικείμενα — χωρίς εγγραφή • Portify

📚 437 βιβλία🎬 22.000+ Ψηφιακά Μαθησιακά Αντικείμενα

Δες τα βιβλία →