Diagonal lines AC and BD intersect at e,

According to cosine theorem, cos

Similarly, cos

∵cos( 180-& lt； BCD)=-cos & lt； BCD =- 1/26 = cos & lt； Not good,

∴<； BAD+& lt； BCD= 180，

∴A, b, c, d four-point * * * circle,

In △ACD and △ABC,

According to cosine theorem,

ac^2=ad^2+cd^2-2ad*cdcos<； adc=ab^2+bc^2-2ab*bc*cos<； ABC，

∵ADC+←ABC = 180，

∴cos<； ABC =-cos & lt； ADC，

169+ 169-2 * 13 * 13 cos & lt； ADC = 49+64+2 * 7 * 8 * cos & lt； ADC，

cos & ltADC= 1/2，

∴<； ADC=60，

AD = CD，

△ ADC is positive △,

Make a positive △ADC circumscribed circle, < ECB = < ADB, (the circumference angle of the same arc is equal),

< Abd = < ACD = 60 (the circumferential angle of the same arc is equal),

< DBC = < DAC = 60° (the circumferential angle of the same arc is equal).

∴〈EBC=〈ABD=60，

∴△ABD∽△EBC，

BC/BD=EC/AD，

8/ 15=OC/ 13，

∴ec=8* 13/ 15= 104/ 15，

CM=AC/2, (n is the midpoint of BD),

EM = EC-AC/2 = 104/ 15- 13/2 = 13/30，

∫△ADE∽△BCE, (the proof method of the symphony theorem),

∴BC/AD=EC/DE，

8/ 13 =( 104/ 15)/DE，

∴DE= 169/ 15，

BE = BD-DE = 15- 169/ 15 = 56/ 15，

EN = ED-BD/2 = 169/ 15- 15/2 = 1 13/30，

∵△ABC∽△DEC, (previously proved),

∴〈BEC=〈BAD，

∴cos<； BEC = cos & lt； BAD=- 1/26，

∵DEC = 180-〈BEC、

∴cos<； DEC= 1/26, (cos< male)

Sin< dec = √ [1-(cos <; MON)^2]= 15√3/26，

In △MEN, according to the cosine theorem,

MN=√57/2，

∵ABC =〈OBD = 120，

< ADB = < ACB, (the circumference angle of the same arc is equal),

∴△ABC∽OBD，

OD/AC=BD/BC，

OD/ 13= 15/8，

OD= 195/8，

OA = OD-AD = 195/8- 13 = 9 1/8，

∫< ABO = < ODC, (the outer angle of a quadrilateral inscribed in a circle is equal to the inner diagonal),

AOB = COD，

∴△OAB∽△OCD，

∴AB/CD=OB/OD，

7/ 13=OB/( 195/8)，

∴OB= 105/8，

OC=OB+BC= 169/8，

Let the distance from O to MN be OH,

Put it through, om,

In △ODB, according to the midline theorem,

on^2=(od^2+ob^2-bd^2/2)/2=( 195^2/64+ 105^2/64- 15^2/2)/2=20925/64=326.953 1，

At △OAC, according to the median line theorem,

om^2=(oa^2+oc^2-ac^2/2)/2=(9 1^2/64+ 169^2/64- 13^2/2)/2= 157 17/64=245.578

According to Pythagorean theorem,

ON^2-(MN+MH)^2=OH^2=OM^2-MH^2，

ON^2-OM^2=(MN+2MH)*MN，

MH=8.8909，

OH=√(OM^2-MH^2)= 12.9046，

The distance from O to MN is about 12.9046.