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When they were shot to death two months ago last Thursday, Richard and Nancy Langert were described by friends and relatives as an All-American couple who had the perfect marriage.

With good jobs, a new home and a first baby on the way, their young lives appeared full of hope and possibility, which made their brutal slayings in the basement of a Winnetka townhouse seem all the more tragic and shocking.

Below the surface, however, the lives of Richard, 30, and Nancy, 25, were far less idyllic. Their relationship had been tinged with indiscretion, and police have discovered several complexities and an intriguing amount of violence on the periphery of their small world.

Interviews with people questioned by police and an examination of public records provide a glimpse of where investigators have looked and what they have discovered while trying to solve the first double murder in Winnetka in this century.

Investigators have traveled to a half-dozen states, examined numerous phone records and other documents and interviewed dozens of people.

They are concentrating on four areas-the couple`s personal lives, the coffee company where they worked, the political activities of Nancy`s sister and the business dealings of her father, attorney M. Lee Bishop. They have ruled out burglary as a motive.

”Many of the people involved in this case are interesting in themselves,” Winnetka Police Chief Herbert Timm said. ”These connections have made it much more fascinating and much more complex than you would expect in a case of this kind.”

In some ways, the killings had the earmarks of a professional hit. Richard had been handcuffed, then shot execution style with a .38 caliber revolver. But in other ways it had been sloppy.

The intruders, who had broken into the townhouse and were waiting for the Langerts when they returned home on the night of April 7, left behind the handcuffs and dropped a glove outside the home while fleeing, apparently over the back fence.

And before she died, Nancy tried to write a message in her blood on the bottom of a knocked-over metal shelf. Though the message is hard to make out, it appears to be ”(H) u.”

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In their search for answers, police have examined the business activities of Lee Bishop, who discovered the bodies the day after the slayings. When he phoned Winnetka police to report the crime, Bishop was crying hysterically.

Investigators discovered that, about a month before the killings, relatives of one of Bishop`s clients were savagely beaten by intruders.

On the afternoon of March 8, two men forced their way into the Palatine home of a pizzeria owner, then pistol-whipped his wife and mother, according to Palatine police. Both women were treated at a hospital for head wounds.

The two men did not appear interested in robbery, police said, and took about $500 only after one of the women offered to pay them to stop the attack. The intruders, who have not been caught, left other valuables behind.

Cook County records show that the pizzeria owner hired Bishop to help him buy a $430,000 apartment building. The man, who is also a bookie, owed Bishop several hundred dollars, which Bishop said was for unpaid legal fees.

The pizzeria owner is not cooperating fully with police investigating either the break-in at his own home or the Langert murders, investigators said.

Police have also questioned one of Bishop`s partners in a recently created recycling and garbage company, as well as a consultant hired by the firm. The two men joined Bishop at the scene of the murders after he telephoned police.

Bishop, who helped set up the company, had been given an interest in it in lieu of legal fees because his new partner was in difficult financial straits.

The partner has had his ups and downs as a businessman, court records show. During the last few months, two banks filed suit attempting to foreclose on his Winnetka home, and on April 3 the Internal Revenue Service filed an $18,000 lien against him.

The consultant, a Park Ridge man, also has had financial problems, filing for personal bankruptcy in August, records show.

And the consultant was involved in a similar recycling operation in Chicago that was temporarily shut down after Chicago sued it for illegally dumping waste.

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Another place where police have looked for leads is the suburban coffee company that employed both Langerts. The owners, who offered a $10,000 reward for information leading to a conviction in the killings, have been cooperating with police.

Investigators quickly dismissed speculation that there might have been a drug connection through the firm. Although it sells coffee from Colombia and other countries, it does not import coffee.

The owners became close to the Langerts after Nancy began working there in the spring of 1989 and Richard was hired to work in a warehouse six months later.

”It was more of a mother-father, daughter-son relationship,” one of the owners said.

The Langerts were in debt and could not afford to buy a house of their own, so the coffee company owners offered to rent them a house they owned in Prospect Heights, with an option to buy it later. The Langerts planned to move there May 1, after living temporarily in the townhouse, owned by Nancy`s parents.

Police were particularly interested in reports that the owner, a gun enthusiast, and his brother-in-law, who also worked at the company, once waved handguns during a confrontation with an employee they suspected of stealing from the firm. The brother-in-law left the company in March.

Police are still trying to question the employee who was threatened.

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One theory that police continue to pursue is that terrorists might have wanted to kidnap Nancy and use her as a bargaining chip with her sister, Jeanne Bishop, who has been active for several years on behalf of Catholics in Northern Ireland.

Although the Irish Republican Army and other Irish groups have never been known to stage a killing in the United States, some investigators believe that this angle could be their best bet.

The Langerts had spent the evening of their murders with Nancy`s sister and parents, celebrating Lee Bishop`s 60th birthday at a chic Italian restaurant in Chicago.

Although the Bishops are Protestants, Jeanne Bishop, a lawyer with a Loop firm, has been active on behalf of the Catholics in Northern Ireland for several years-joining groups, writing articles and traveling several times to England and Ireland to publicize the plight of Ulster Catholics.

Jeanne Bishop also became an advocate for Joseph Doherty, jailed on murder charges in New York City for seven years while fighting extradition to England. During one of her visits to Doherty, Jeanne took along Nancy and their mother.

Richard and Nancy traveled with her sister and parents to England and Scotland in 1988. Nancy told a friend she was nervous during the trip and thought about drawing up a will.

Last October, the FBI notified Jeanne Bishop that an informant had told them of a threat on her life related to her political activities. She dismissed the threat because the FBI would not tell her the source, calling it a ploy to get her to reveal her Irish contacts.

In attempting to determine whether a terrorist group might have been linked to the Langert killings, police have traveled extensively: to New Hampshire, to interview Jeanne Bishop`s ex-husband; to New York, in an unsuccessful attempt to speak with Doherty; and to Pennsylvania, to interview a woman who accompanied Jeanne Bishop on her most recent trip to Northern Ireland, which ended three days before the Langerts were slain.

Police say they have been hampered by the continued unwillingness of Jeanne Bishop and other members of her family to discuss her human rights activities.

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Police have also focused on the lifestyles of the two victims. The couple`s marriage was not always trouble-free. In 1987, shortly after they were wed, Nancy became involved with another man.

”The biggest problem was Rich always worked nights, that they were never together,” a friend said.

Later Nancy would blame the affair on immaturity. Friends say she was very committed to Richard at the time of their deaths.

Police have found no indication of any other infidelities and ruled that out as a factor in the killings after interviewing Nancy`s former boyfriend, who now lives in the Southeast.

But a typewritten, 6-year-old lovers` agreement drawn up by Nancy and signed by both of them shortly after they met was lying on the couple`s kitchen table when they were slain. It remains a mystery why the couple might have been concerned with the agreement at the time.

The agreement listed the high qualities the two would have to possess if their relationship was to thrive: patience, understanding, trust and love.

Although they were earning a combined $60,000 at the time of their deaths, the Langerts never could quite get their finances together. They were living from paycheck to paycheck and often had credit card debts. They left a $10,000 insurance policy.

But police have not found anything unusual in their finances. ”There seems to be nothing in their personal lives that would have caused them to be the victims of murder,” Timm said.

The special 9-person task force set up to investigate the killings, with officers from four nearby suburbs, was disbanded at the beginning of June. But five Winnetka officers will continue to focus on the case, Timm said.

”We are committed to keeping the case active,” he said.