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Category: Geometry

Let-ABCD-be-a-convex-quadrilateral-with-DAB-BDC-90-Let-the-incircles-of-triangles-ABD-and-BCD-touch-BD-at-P-and-Q-respectively-with-P-lying-in-between-B-and-Q-If-AD-999-and-PQ-200-then-

Question Number 19792 by Tinkutara last updated on 15/Aug/17 $$\mathrm{Let}\:{ABCD}\:\mathrm{be}\:\mathrm{a}\:\mathrm{convex}\:\mathrm{quadrilateral} \\ $$$$\mathrm{with}\:\angle{DAB}\:=\:\angle{BDC}\:=\:\mathrm{90}°.\:\mathrm{Let}\:\mathrm{the} \\ $$$$\mathrm{incircles}\:\mathrm{of}\:\mathrm{triangles}\:{ABD}\:\mathrm{and}\:{BCD} \\ $$$$\mathrm{touch}\:{BD}\:\mathrm{at}\:{P}\:\mathrm{and}\:{Q},\:\mathrm{respectively}, \\ $$$$\mathrm{with}\:{P}\:\mathrm{lying}\:\mathrm{in}\:\mathrm{between}\:{B}\:\mathrm{and}\:{Q}.\:\mathrm{If} \\ $$$${AD}\:=\:\mathrm{999}\:\mathrm{and}\:{PQ}\:=\:\mathrm{200}\:\mathrm{then}\:\mathrm{what}\:\mathrm{is} \\ $$$$\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{the}\:\mathrm{radii}\:\mathrm{of}\:\mathrm{the}\:\mathrm{incircles}\:\mathrm{of} \\ $$$$\mathrm{triangles}\:{ABD}\:\mathrm{and}\:{BDC}? \\…

In-a-triangle-ABC-with-BCA-90-the-perpendicular-bisector-of-AB-intersects-segments-AB-and-AC-at-X-and-Y-respectively-If-the-ratio-of-the-area-of-quadrilateral-BXYC-to-the-area-of-triangle-ABC-i

Question Number 19794 by Tinkutara last updated on 15/Aug/17 $$\mathrm{In}\:\mathrm{a}\:\mathrm{triangle}\:{ABC}\:\mathrm{with}\:\angle{BCA}\:=\:\mathrm{90}°, \\ $$$$\mathrm{the}\:\mathrm{perpendicular}\:\mathrm{bisector}\:\mathrm{of}\:{AB} \\ $$$$\mathrm{intersects}\:\mathrm{segments}\:{AB}\:\mathrm{and}\:{AC}\:\mathrm{at}\:{X} \\ $$$$\mathrm{and}\:{Y},\:\mathrm{respectively}.\:\mathrm{If}\:\mathrm{the}\:\mathrm{ratio}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{area}\:\mathrm{of}\:\mathrm{quadrilateral}\:{BXYC}\:\mathrm{to}\:\mathrm{the} \\ $$$$\mathrm{area}\:\mathrm{of}\:\mathrm{triangle}\:{ABC}\:\mathrm{is}\:\mathrm{13}\::\:\mathrm{18}\:\mathrm{and} \\ $$$${BC}\:=\:\mathrm{12}\:\mathrm{then}\:\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{length}\:\mathrm{of}\:{AC}? \\ $$ Terms…

The-sides-of-a-triangle-are-of-lengths-m-2-n-2-m-2-n-2-2mn-Show-that-it-is-a-right-angle-

Question Number 19783 by NECC last updated on 15/Aug/17 $${The}\:{sides}\:{of}\:{a}\:{triangle}\:{are}\:{of} \\ $$$${lengths}\:\sqrt{\left({m}^{\mathrm{2}} −{n}^{\mathrm{2}} \right)}\:,{m}^{\mathrm{2}} +{n}^{\mathrm{2}} ,\:\mathrm{2}{mn}. \\ $$$${Show}\:{that}\:{it}\:{is}\:{a}\:{right}\:{angle}\:\Delta. \\ $$ Commented by Tinkutara last updated…