Courses / Mathematics / Geometric constructions, circle proofs and coordinate loci Mathematics · unit version 1 Geometric constructions, circle proofs and coordinate loci School-entry constructions and coordinate proofs with explicit tools, reversible locus arguments and general-triangle measurement. Public written practice requires review; completion is not an independent mastery claim.
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What must a construction record specify? A plausible sketch is not enough to reproduce or justify a construction.
Compare reasoning and find a refresher Name each center, radius, line, intersection and chosen side or ray. State why the requested intersections exist and why the final object has the desired property.
Review compass-and-straightedge operations
Why does a similarity preserve angle measures? Circle and triangle arguments use transformations as well as calculations.
Compare reasoning and find a refresher A positive dilation preserves ray directions and multiplies all lengths by one factor. Rigid motions preserve lengths and angles. Their composition therefore preserves corresponding angles.
Review similarity witnesses
Can squaring both sides introduce extra solutions? Focal-locus derivations must prove both directions and retain sign restrictions.
Compare reasoning and find a refresher Yes: u=v implies u²=v², but the latter also permits u=−v. Record nonnegative distance and branch conditions before recovering a square root.
Review reversible radical equations
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Construct, prove and measure unfamiliar configurations
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Lesson details and objectives Turn compass-and-straightedge instructions into justified constructions, including angle bisectors and a folding alternative.
Execute a segment and an angle copy without a graduated ruler. Prove the copied angle and its bisector using congruent triangles. Specify tool assumptions and distinguish exact construction from a measured drawing. 6 practice tasks.
Lesson details and objectives Build through-point lines with compass arcs, then justify a second method using folds.
Construct a perpendicular through an on-line or off-line point. Construct a parallel through an external point and prove it is unique. Prove the perpendicular-bisector equidistance locus in both directions. 6 practice tasks.
Lesson details and objectives Construct three regular figures from a given circle and prove their equal sides and angles.
Construct a regular hexagon by stepping the radius around a circle. Construct an inscribed equilateral triangle and square. Justify closure and regularity rather than trusting accumulated compass steps. 6 practice tasks.
Lesson details and objectives Use equidistance loci to construct triangle centers and prove the circles actually touch the intended objects.
Construct a circumcenter with perpendicular bisectors and prove concurrence. Construct an incenter with internal angle bisectors and perpendicular radii. Justify incircle contact on side segments and handle nondegenerate triangle assumptions. 6 practice tasks.
Lesson details and objectives Connect inscribed, cyclic and tangent configurations using named arcs and full angle arguments.
Derive an inscribed-angle rule with correct intercepted arc selection. Prove the opposite-angle theorem for cyclic quadrilaterals. Relate the angle between two tangents to minor and major arcs. 6 practice tasks.
Lesson details and objectives Build a diameter circle to find contact points and prove completeness of the construction.
Construct the two tangent contacts using the circle on diameter OP. Prove each constructed line is tangent and every tangent contact is found. Classify external, on-circle and interior point cases. 6 practice tasks.
Lesson details and objectives Derive a circle locus in both directions and classify completed-square degeneracies.
Construct a similarity between arbitrary positive-radius circles and prove onto equality. Derive and reverse the arbitrary-center circle equation. Complete squares and distinguish positive, zero and negative squared-radius cases. 6 practice tasks.
Lesson details and objectives Use similarity and angular additivity to derive arc and sector formulas, including a school-level limiting argument.
Prove arc length scales with radius under a similarity. Derive radian measure and sector area from circle measure. Keep degrees, radians, lengths and areas distinct in applications. 6 practice tasks.
Lesson details and objectives Derive area and sine relations across acute, right and obtuse triangles, then audit measurement ambiguity.
Construct an altitude to derive the sine area formula for acute and obtuse angles. Derive the sine law with stated angle correspondence. Use sine/cosine laws in measurements and test SSA feasibility and ambiguity. 6 practice tasks.
Lesson details and objectives Derive parallel and perpendicular line rules without losing vertical cases.
Prove finite-slope parallel and perpendicular criteria from direction vectors. Construct the parallel or perpendicular through a specified point. Handle vertical, horizontal and coincident-line exceptions explicitly. 6 practice tasks.
Lesson details and objectives Connect internal ratios to affine coordinates and distinguish extrapolation from division inside a segment.
Derive the internal m:n section formula from a directed fraction. Construct a prescribed rational division using parallel lines. Interpret reversed endpoints and exterior parameters without unsigned-ratio confusion. 6 practice tasks.
Lesson details and objectives Choose useful bases and perpendicular heights for tilted polygons instead of reading vertical coordinate gaps as heights.
Compute non-axis-aligned polygon perimeter from distance. Justify triangle and rectangle area from perpendicular lengths or decomposition. Detect invalid vertex order and distinguish coordinate units from diagram pixels. 6 practice tasks.
Lesson details and objectives Turn the two-focus definition into a standard equation and prove the reverse implication with explicit signs.
Derive an ellipse equation from a focal distance sum. Prove the converse without admitting spurious squared solutions. Classify valid parameters, translated centers and limiting degeneracies. 6 practice tasks.
Lesson details and objectives Use an absolute distance difference to derive two branches and keep the sign restrictions through squaring.
Derive a hyperbola equation from a fixed absolute focal-distance difference. Prove the converse on both branches using the correct distance signs. Classify zero, maximal and impossible distance differences. 6 practice tasks.
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