Summary of "🎓 פתרון בגרות 571 חורף 2026 שאלה 4 -גיאומטריה עם מעגל."
Context and givens
- Solution to Winter 2026 Matriculation Exam, Question 571 (geometry).
- Triangle ABC with angles BAC = BCA = α (isosceles, base angles equal).
- Point K lies on the extension of AB. The line through K and C meets the given circle again at F (F is the second intersection of KC with the circle).
Goals
- Show AF is a diameter of the circle.
- Deduce several angle/length relations and right-angle facts.
- Prove AD = DK.
- Show quadrilateral BDCK is cyclic.
- Derive the relation CK · AF = AD · AC (equivalently AD/AC = CK/AF).
Main ideas, claims and reasoning
Recognize the isosceles triangle
- From BAC = BCA = α conclude triangle ABC is isosceles; mark equal base angles and related equal arcs/sides.
Prove AF is a diameter
- Use the inscribed-angle / diameter theorem (and its converse): an inscribed right angle subtends a diameter.
- Identify that angle ACF (or equivalent) is 90°, hence AF is a diameter of the circle.
Perpendicular / midpoint properties
- Apply the midpoint-of-hypotenuse fact: in a right triangle the midpoint of the hypotenuse is equidistant from all vertices.
- When a median equals half the side it meets, conclude the triangle is right and the median goes to the hypotenuse midpoint; use this to deduce perpendicularity and midpoint relations in the configuration (e.g., DB is perpendicular and B is a midpoint in the relevant triangle).
Prove AD = DK
- Show a segment acts both as a median and an altitude in a relevant triangle (e.g., FB in triangle AFK), which forces isosceles/triangle congruence conclusions.
- Use congruence or equal-distance arguments (triangles sharing DF, equal angles/sides) to obtain AD = DK.
Show BDCK is cyclic
- Use the cyclic-quadrilateral criterion: if one pair of opposite angles sum to 180° then the four points are concyclic.
- Find that one opposite angle is 90° and the opposite angle is also 90°, so their sum is 180° and BDCK is cyclic.
Prove corresponding-angle equality for similarity
- Identify two right triangles (D C K and A C F). Both have a right angle and share a corresponding acute angle (labelled β).
- With right angle + one equal acute angle, the triangles are similar by AA.
Use similarity to get the product relation
- From similarity of triangles DCK and ACF, set up corresponding side ratios.
- Substitute DK = AD (from earlier) to obtain the final product equality CK · AF = AD · AC.
Step-by-step method (followable)
- Mark the given: ∠BAC = ∠BCA = α → triangle ABC is isosceles.
- Note line KC meets the circle again at F; consider chord AF.
- Show ∠ACF = 90° (from the configuration) → AF is a diameter (inscribed-angle/diameter theorem).
- Use the midpoint-of-hypotenuse fact where a median equals half the side it meets to deduce right-triangle midpoint/perpendicular relations (apply to relevant triangles to get DB ⟂ something and midpoint claims).
- Show a segment is both median and altitude in triangle AFK (FB in the speaker’s explanation) → triangle AFK is isosceles and base angles equal.
- Prove AD = DK using triangle congruence arguments (e.g., triangles with common DF and equal angles/sides).
- Observe that quadrilateral BDCK has opposite angles summing to 180° (both 90°) → BDCK is cyclic.
- Identify triangles DCK and ACF as right and share an acute angle β → triangles are similar by AA.
- From similarity, write corresponding side ratios and replace KD by AD to get CK · AF = AD · AC.
- Conclude the desired relation and finish.
Theorems and lemmas used
- In an isosceles triangle, base angles opposite equal sides are equal.
- Inscribed-angle theorem and its converse (an inscribed right angle subtends a diameter).
- Midpoint-of-hypotenuse theorem: the midpoint of the hypotenuse of a right triangle is equidistant from the three vertices.
- If the median and altitude from the same vertex coincide, the triangle is isosceles (legs equal).
- A quadrilateral is cyclic iff a pair of opposite angles sum to 180°.
- Similar triangles: AA criterion and the corresponding-side proportionality.
Final result
CK · AF = AD · AC (equivalently AD / AC = CK / AF)
Sources
- Problem: Winter 2026 Matriculation Examination, Question 571 (geometry).
- Solution as presented by the (unnamed) video presenter/solver.
Category
Educational
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